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These numbers are the results of finding combinations of <em>n</em> things taken <em>k</em> at a time. For quick reference, the first ten rows of the triangle are shown.</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-252745\" src=\"https://www.dummies.com/wp-content/uploads/finitemath-pascals.jpg\" alt=\"finitemath-pascals\" width=\"433\" height=\"200\" /></p>\n"},{"title":"Binomial distributions","thumb":null,"image":null,"content":"<p>A common situation when doing probability problems is having to determine the patterns of heads-and-tails, boys-and-girls, true-or-false arrangements. When there are two choices, there are 2<em><sup>n</sup></em> ways that they can occur. The following refer to Heads-or-Tails in a coin-flipping situation but can be adapted to any binomial arrangement.</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-252749\" src=\"https://www.dummies.com/wp-content/uploads/finitemath-one-toss.jpg\" alt=\"finitemath-one-toss\" width=\"535\" height=\"989\" /></p>\n"},{"title":"Matrix notation","thumb":null,"image":null,"content":"<p>Matrices are rectangular arrangements of elements. The dimension of a matrix is given with <em>m</em> × <em>n</em> where <em>m</em> is the number of rows and <em>n </em>is the number of columns. The elements are identified with subscripts giving the row, <em>j</em>, and column,<em> k</em>, shown as <em>a<sub>jk</sub></em> for the elements of a matrix A.</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-252754\" src=\"https://www.dummies.com/wp-content/uploads/Matrix-A.jpg\" alt=\"Matrix-A\" width=\"535\" height=\"335\" /></p>\n<p>When multiplying the matrices, the number of rows in the first matrix has to equal the number of columns in the second. Given matrices A and B where A has dimension 2 × 3 and B has the dimension3 × 2, the resulting matrices are found as follows:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-252755\" src=\"https://www.dummies.com/wp-content/uploads/Matrix-B.jpg\" alt=\"Matrix-B\" width=\"535\" height=\"191\" /></p>\n"},{"title":"Factorial","thumb":null,"image":null,"content":"<p>The factorial operation says to multiply the designated number by every positive integer smaller than that number.</p>\n<p><em>n</em>! = <em>n</em> (<em>n</em> – 1) (<em>n</em> – 2)   3 2 1</p>\n<p>When using the operation in the formulas for the number of permutations or combinations of <em>n</em> things taken <em>k</em> at a time, factorial values need to be inserted into the numerator and denominator of the fraction. The first sixteen factorial values are given here. And, by definition, 0! = 1.</p>\n<table>\n<tbody>\n<tr>\n<td width=\"53\"><em>n</em></td>\n<td width=\"108\"><em>n</em>!</td>\n<td width=\"60\"><em>n</em></td>\n<td width=\"168\"><em>n</em>!</td>\n</tr>\n<tr>\n<td width=\"53\">1</td>\n<td width=\"108\">1</td>\n<td width=\"60\">9</td>\n<td width=\"168\">362,880</td>\n</tr>\n<tr>\n<td width=\"53\">2</td>\n<td width=\"108\">2</td>\n<td width=\"60\">10</td>\n<td width=\"168\">3,628,800</td>\n</tr>\n<tr>\n<td width=\"53\">3</td>\n<td width=\"108\">6</td>\n<td width=\"60\">11</td>\n<td width=\"168\">39,916,800</td>\n</tr>\n<tr>\n<td width=\"53\">4</td>\n<td width=\"108\">24</td>\n<td width=\"60\">12</td>\n<td width=\"168\">479,001,600</td>\n</tr>\n<tr>\n<td width=\"53\">5</td>\n<td width=\"108\">120</td>\n<td width=\"60\">13</td>\n<td width=\"168\">6,227,020,800</td>\n</tr>\n<tr>\n<td width=\"53\">6</td>\n<td width=\"108\">720</td>\n<td width=\"60\">14</td>\n<td width=\"168\">87,178,291,200</td>\n</tr>\n<tr>\n<td width=\"53\">7</td>\n<td width=\"108\">5,040</td>\n<td width=\"60\">15</td>\n<td width=\"168\">1,307,674,368,000</td>\n</tr>\n<tr>\n<td width=\"53\">8</td>\n<td width=\"108\">40,320</td>\n<td width=\"60\">16</td>\n<td width=\"168\">20,922,789,888,000</td>\n</tr>\n</tbody>\n</table>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-03-08T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":252761},{"headers":{"creationTime":"2016-03-27T16:48:32+00:00","modifiedTime":"2022-02-16T20:29:06+00:00","timestamp":"2022-02-24T17:07:33+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"Reading Financial Reports For Dummies Cheat Sheet","strippedTitle":"reading financial reports for dummies cheat sheet","slug":"reading-financial-reports-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"This Cheat Sheet includes essentials of reading financial reports, including annual reports, SEC reports, profitability ratios, and more.","noIndex":0,"noFollow":0},"content":"If you're looking at a business with an interest in investing in it, you need to read its financial reports. Of course, when it comes to the annual report, you don't need to read everything, just the key parts.\r\n\r\nCombining the annual report with some of the financial reports a corporation files with the Securities and Exchange Commission (SEC) can help you figure profitability and liquidity ratios and get a better sense of cash flow.\r\n\r\nKeep this handy Cheat Sheet nearby for a quick reference to reading financial reports, including SEC reports, profitability ratios, liquidity ratios, and cash flow formulas.","description":"If you're looking at a business with an interest in investing in it, you need to read its financial reports. Of course, when it comes to the annual report, you don't need to read everything, just the key parts.\r\n\r\nCombining the annual report with some of the financial reports a corporation files with the Securities and Exchange Commission (SEC) can help you figure profitability and liquidity ratios and get a better sense of cash flow.\r\n\r\nKeep this handy Cheat Sheet nearby for a quick reference to reading financial reports, including SEC reports, profitability ratios, liquidity ratios, and cash flow formulas.","blurb":"","authors":[{"authorId":8974,"name":"Lita Epstein","slug":"lita-epstein","description":"Lita Epstein, MBA, enjoys helping people develop good financial, investing, and tax planning skills. 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She designs and teaches online courses and has written more than 40 books, including Bookkeeping For Dummies and Reading Financial Reports For Dummies.</p>","authors":[{"authorId":8974,"name":"Lita Epstein","slug":"lita-epstein","description":"Lita Epstein, MBA, enjoys helping people develop good financial, investing, and tax planning skills. She designs and teaches online courses and has written more than 40 books, including Bookkeeping For Dummies and Reading Financial Reports For Dummies.","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8974"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119871361&quot;]}]\" id=\"du-slot-6217bb5589827\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119871361&quot;]}]\" id=\"du-slot-6217bb558a1c3\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":159936,"title":"Key Parts in an Annual Report","slug":"key-parts-in-an-annual-report","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/159936"}},{"articleId":159926,"title":"Key Securities and Exchange Commission Reports","slug":"key-securities-and-exchange-commission-reports","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/159926"}},{"articleId":159954,"title":"Reading Financial Reports for Profitability Ratios","slug":"reading-financial-reports-for-profitability-ratios","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/159954"}},{"articleId":159924,"title":"Reading Financial Reports for Liquidity Ratios","slug":"reading-financial-reports-for-liquidity-ratios","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/159924"}},{"articleId":159925,"title":"Reading Financial Reports for Cash Flow Formulas","slug":"reading-financial-reports-for-cash-flow-formulas","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/159925"}}],"content":[{"title":"Key parts in an annual report","thumb":null,"image":null,"content":"<p>Annual reports can be daunting, and you may be relieved to know that you don’t actually need to scour every page of one. The following parts best serve to give you the big picture:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><b>Auditor’s report:</b> Tells you whether the numbers are accurate and whether you should have any concerns about the future operation of the business</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Financial statements:</b> The balance sheet, the income statement, and the statement of cash flows; where you find the actual financial results for the year</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Notes to the financial statements:</b> Details about potential problems with the numbers or how the numbers were derived</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Management’s discussion and analysis:</b> The higher-ups’ breakdown of the financial results and other factors that impact the company’s operations</p>\n</li>\n</ul>\n<p>The rest is fluff.</p>\n"},{"title":"Key Securities and Exchange Commission Reports","thumb":null,"image":null,"content":"<p>Reports to the government are more extensive than the glossy reports sent to shareholders. Although many different types of forms must be filed with the Securities and Exchange Commission, you can get most of the juicy information from just a few:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><b>10-K:</b> Annual report that provides a comprehensive overview of the corporation’s business</p>\n</li>\n<li>\n<p class=\"first-para\"><b>10-Q:</b> Quarterly report that describes key financial information about the prior three months</p>\n</li>\n<li>\n<p class=\"first-para\"><b>8-K:</b> Shows any major events that could impact the financial position of the company</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Forms 3 to 5:</b> Reflect changes in ownership of stock by directors, officers, and major stockholders, giving you an idea of the view from the inside</p>\n</li>\n</ul>\n"},{"title":"Profitability ratios","thumb":null,"image":null,"content":"<p>You read financial reports to get a sense of a company&#8217;s financial position and how viable it is in the marketplace. You can test a company&#8217;s money-making prowess using the following important formulas.</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><i>Price/earnings ratio </i>compares the price of a stock to its earnings. A ratio of 10 means that for every $1 in company earnings per share, people are willing to pay $10 per share to buy the stock.</p>\n<p class=\"child-para\">Price/earnings ratio = Market value per share of stock divided by Earnings per share of stock</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Dividend payout ratio</i> shows the amount of a company&#8217;s earnings that are paid out to investors. Use it to determine the actual cash return you get by buying and holding a share of stock.</p>\n<p class=\"child-para\">Dividend payout ratio = Yearly dividend per share divided by Earnings per share</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Return on sales</i> tests how efficiently a company is running its operations by measuring the profit produced per dollar of sales.</p>\n<p class=\"child-para\">Return on sales = Net income before taxes divided by Sales</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Return on assets </i>shows you how well a company uses its assets. A high return on assets usually means the company is managing its assets well.</p>\n<p class=\"child-para\">Return on assets = Net income divided by Total assets</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Return on equity</i> measures how well a company earns money for its investors.</p>\n<p class=\"child-para\">Return on equity = Net income divided by Shareholders&#8217; equity</p>\n</li>\n<li>\n<p class=\"first-para\">The <i>gross margin</i> gives you a picture of how much revenue is left after all the direct costs of producing and selling the product have been subtracted.</p>\n<p class=\"child-para\">Gross margin = Gross profit divided by Net sales or revenues</p>\n</li>\n<li>\n<p class=\"first-para\">The <i>operating margin</i> looks at how well a company controls costs, factoring in any expenses not directly related to the production and sales of a particular product.</p>\n<p class=\"child-para\">Operating margin = Operating profit divided by Net sales or revenues</p>\n</li>\n</ul>\n"},{"title":"Liquidity ratios","thumb":null,"image":null,"content":"<p>If a company doesn’t have cash on hand to cover its day-to-day operations, it’s probably on shaky ground. Use the following formulas to find out whether a company has plenty of liquid (easily converted to cash) assets.</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><i>Current ratio</i> gives you a good idea of whether a company will be able to pay any bills due over the next 12 months with assets it has on hand.</p>\n<p class=\"child-para\">Current ratio = Current assets divided by Current liabilities</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Quick ratio</i> or <i>acid test ratio</i> shows a company’s ability to pay its bills using only cash on hand or cash already due from accounts receivable. It doesn’t include money anticipated from the sale of inventory and the collection of the money from those sales.</p>\n<p class=\"child-para\">Quick ratio = Quick assets divided by Current liabilities</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Interest coverage ratio</i> lets you know whether a company is bringing in enough money to pay interest on whatever outstanding debt it has.</p>\n<p class=\"child-para\">Interest coverage ratio = EBITDA divided by Interest expense</p>\n</li>\n</ul>\n"},{"title":"Cash flow formulas","thumb":null,"image":null,"content":"<p>You&#8217;re interested in a company, so you&#8217;re reading its financial reports. Part of the test of a viable operation is having enough cash to keep the company going. Use the following formulas to make sure a company has plenty of cash to keep operating.</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><i>Free cash flow</i> shows you how much money a company earns from its operations that can actually be put in a savings account for future use.</p>\n<p class=\"child-para\">Free cash flow = Cash provided by operating activities – Capital expenditures – Cash dividends</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Cash return on sales</i> looks specifically at how much cash is being generated by sales.</p>\n<p class=\"child-para\">Cash return on sales = Cash provided by operating activities divided by Net sales</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Current cash debt coverage ratio </i>lets you know whether a company has enough cash to meet its short-term needs.</p>\n<p class=\"child-para\">Current cash debt coverage ratio = Cash provided by operating activities divided by Average current liabilities</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Cash flow coverage ratio</i> finds out whether a company has enough money to cover its bills <i>and</i> finance growth.</p>\n<p class=\"child-para\">Cash flow coverage ratio = Cash flows from operating activities divided by Cash requirements</p>\n</li>\n</ul>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Two years","lifeExpectancySetFrom":"2022-02-07T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":207792},{"headers":{"creationTime":"2016-03-27T16:48:44+00:00","modifiedTime":"2022-01-31T19:37:09+00:00","timestamp":"2022-02-24T17:07:27+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"Business Statistics For Dummies Cheat Sheet","strippedTitle":"business statistics for dummies cheat sheet","slug":"business-statistics-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"This Cheat Sheet is a handy resource for the basics of business statistics, including hypothesis testing, sampling distributions, and more.","noIndex":0,"noFollow":0},"content":"Statistics make it possible to analyze real-world business problems with actual data so that you can determine if a marketing strategy is really working, how much a company should charge for its products, or any of a million other practical questions. The science of statistics uses regression analysis, hypothesis testing, sampling distributions, and more to ensure accurate data analysis.","description":"Statistics make it possible to analyze real-world business problems with actual data so that you can determine if a marketing strategy is really working, how much a company should charge for its products, or any of a million other practical questions. The science of statistics uses regression analysis, hypothesis testing, sampling distributions, and more to ensure accurate data analysis.","blurb":"","authors":[{"authorId":9080,"name":"Alan Anderson","slug":"alan-anderson","description":"Alan Anderson, PhD, is a professor of economics and finance at Fordham University and New York University. He's a veteran economist, risk manager, and fixed income analyst. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/9080"}}],"primaryCategoryTaxonomy":{"categoryId":34229,"title":"Calculation & Analysis","slug":"calculation-analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":162083,"title":"How Businesses Use Regression Analysis Statistics","slug":"how-businesses-use-regression-analysis-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162083"}},{"articleId":162074,"title":"Random Variables and Probability Distributions in Business Statistics","slug":"random-variables-and-probability-distributions-in-business-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162074"}},{"articleId":162073,"title":"Explore Hypothesis Testing in Business Statistics","slug":"explore-hypothesis-testing-in-business-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162073"}},{"articleId":162066,"title":"3 Ways to Describe Populations and Samples in Business Statistics","slug":"3-ways-to-describe-populations-and-samples-in-business-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162066"}},{"articleId":162062,"title":"Business Statistics: Assess Financial Risk with the VaR Methodology","slug":"business-statistics-assess-financial-risk-with-the-var-methodology","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162062"}}],"fromCategory":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282040,"slug":"business-statistics-for-dummies","isbn":"9781118630693","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"amazon":{"default":"https://www.amazon.com/gp/product/1118630696/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1118630696/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1118630696-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1118630696/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1118630696/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/business-statistics-for-dummies-cover-9781118630693-203x255.jpg","width":203,"height":255},"title":"Business Statistics For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <p><b data-author-id=\"9080\">Alan Anderson</b>, PhD is a teacher of finance, economics, statistics, and math at Fordham and Fairfield universities as well as at Manhattanville and Purchase colleges. Outside of the academic environment he has many years of experience working as an economist, risk manager, and fixed income analyst. Alan received his PhD in economics from Fordham University, and an M.S. in financial engineering from Polytechnic University.</p>","authors":[{"authorId":9080,"name":"Alan Anderson","slug":"alan-anderson","description":"Alan Anderson, PhD, is a professor of economics and finance at Fordham University and New York University. He's a veteran economist, risk manager, and fixed income analyst. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/9080"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118630693&quot;]}]\" id=\"du-slot-6217bb4fb6328\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118630693&quot;]}]\" id=\"du-slot-6217bb4fb6d08\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":162066,"title":"3 Ways to Describe Populations and Samples in Business Statistics","slug":"3-ways-to-describe-populations-and-samples-in-business-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162066"}},{"articleId":162074,"title":"Random Variables and Probability Distributions in Business Statistics","slug":"random-variables-and-probability-distributions-in-business-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162074"}},{"articleId":162063,"title":"Understand Sampling Distributions in Business Statistics","slug":"understand-sampling-distributions-in-business-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162063"}},{"articleId":162073,"title":"Explore Hypothesis Testing in Business Statistics","slug":"explore-hypothesis-testing-in-business-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162073"}},{"articleId":162083,"title":"How Businesses Use Regression Analysis Statistics","slug":"how-businesses-use-regression-analysis-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162083"}}],"content":[{"title":"3 ways to describe populations and samples","thumb":null,"image":null,"content":"<p>When you&#8217;re working with populations and samples (a subset of a population) in business statistics, you can use three common types of measures to describe the data set: central tendency, dispersion, and association.</p>\n<p class=\"Tip\">By convention, the statistical formulas used to describe population measures contain Greek letters, while the formulas used to describe sample measures contain Latin letters.</p>\n<h3>Measures of central tendency</h3>\n<p>In statistics, the mean, median, and mode are known as measures of <em>central tendency</em>; they are used to identify the center of a data set:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><strong>Mean:</strong> The value between the largest and smallest values of a data set, obtained by a prescribed method.</p>\n</li>\n<li>\n<p class=\"first-para\"><strong>Median: </strong>The value which divides a data set into two equal halves</p>\n</li>\n<li>\n<p class=\"first-para\"><strong>Mode:</strong> The most commonly observed value in a data set</p>\n</li>\n</ul>\n<p>Samples are randomly chosen from populations. If this process is carried out correctly, each sample should accurately reflect the characteristics of the population. So, a sample measure, such as the <em>mean,</em> should be a good estimate of the corresponding population measure. Consider the following examples of mean:</p>\n<p>Population mean:</p>\n<p>This formula simply tells you to add up all the elements in the population and divide by the size of the population.</p>\n<p>Sample mean:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393293.image0.jpg\" alt=\"image0.jpg\" width=\"106\" height=\"93\" /></p>\n<p>The process for computing this is exactly the same; you add up all the elements in the sample and divide by the size of the sample.</p>\n<p>In addition to measures of central tendency, two other key types of measures are measures of dispersion (spread) and measures of association.</p>\n<h3>Measures of dispersion</h3>\n<p><em>M</em><em>easures of dispersion</em> include variance/standard deviation and percentiles/quartiles/interquartile range. The variance and standard deviation are closely related to each other; the standard deviation always equals the <em>square root</em> of the variance.</p>\n<p>The formulas for the population and sample variance are:</p>\n<p>Population variance:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393294.image1.jpg\" alt=\"image1.jpg\" width=\"159\" height=\"86\" /></p>\n<p>Sample variance:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393295.image2.jpg\" alt=\"image2.jpg\" width=\"159\" height=\"86\" /></p>\n<p class=\"Remember\"><em>Percentiles</em> split up a data set into 100 equal parts each consisting of 1 percent of the values in the data set. <em>Quartiles</em> are a special type of percentiles; they split up the data into four equal parts. The <em>interquartile</em> range represents the middle 50 percent of the data; it&#8217;s calculated as the third quartile minus the first quartile.</p>\n<h3>Measures of association</h3>\n<p>Another type of measure, known as a <em>measure of </em><em>association</em>, refers to the <em>relationship </em>between two samples or two populations. Two examples of this are the <em>covariance</em> and the <em>correlation</em>:</p>\n<p>Population covariance:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393296.image3.jpg\" alt=\"image3.jpg\" width=\"257\" height=\"86\" /></p>\n<p>Sample covariance:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393297.image4.jpg\" alt=\"image4.jpg\" width=\"257\" height=\"86\" /></p>\n<p>Population correlation:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393298.image5.jpg\" alt=\"image5.jpg\" width=\"127\" height=\"57\" /></p>\n<p>Sample correlation:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393299.image6.jpg\" alt=\"image6.jpg\" width=\"110\" height=\"61\" /></p>\n<p>The correlation is closely related to the covariance; it&#8217;s defined to ensure that its value is always between negative one and positive one.</p>\n"},{"title":"Random variables and probability distributions","thumb":null,"image":null,"content":"<p>Random variables and probability distributions are two of the most important concepts in statistics. A <i>random variable</i> assigns unique numerical values to the outcomes of a <i>random experiment; </i>this is a process that generates uncertain outcomes. A <i>probability distribution</i> assigns probabilities to each possible value of a random variable.</p>\n<p>The two basic types of probability distributions are discrete and continuous. A discrete probability distribution can only assume a <i>finite</i> number of different values.</p>\n<p>Examples of discrete distributions include:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Binomial</p>\n</li>\n<li>\n<p class=\"first-para\">Geometric</p>\n</li>\n<li>\n<p class=\"first-para\">Poisson</p>\n</li>\n</ul>\n<p>A continuous probability distribution can assume an <i>infinite</i> number of different values. Examples of continuous distributions include:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Uniform</p>\n</li>\n<li>\n<p class=\"first-para\">Normal</p>\n</li>\n<li>\n<p class=\"first-para\">Student&#8217;s t</p>\n</li>\n<li>\n<p class=\"first-para\">Chi-square</p>\n</li>\n<li>\n<p class=\"first-para\">F</p>\n</li>\n</ul>\n"},{"title":"Understand sampling distributions","thumb":null,"image":null,"content":"<p>In statistics, sampling distributions are the probability distributions of any given statistic based on a random sample, and are important because they provide a major simplification on the route to statistical inference. More specifically, they allow analytical considerations to be based on the sampling distribution of a statistic, rather than on the joint probability distribution of all the individual sample values.</p>\n<p class=\"Remember\">The value of a sample statistic such as the sample mean (X) is likely to be different for each sample that is drawn from a population. It can, therefore, be thought of as a <i>random variable</i>, whose properties can be described with a <i>probability distribution</i>. The probability distribution of a sample statistic is known as a <i>sampling distribution</i>.</p>\n<p>According to a key result in statistics known as the Central Limit Theorem, the sampling distribution of the sample mean is normal if one of two things is true:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">The underlying population is normal</p>\n</li>\n<li>\n<p class=\"first-para\">The sample size is at least 30</p>\n</li>\n</ul>\n<p>Two moments are needed to compute probabilities for the sample mean; the mean of the sampling distribution equals:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393317.image0.png\" alt=\"image0.png\" width=\"63\" height=\"32\" /></p>\n<p>The standard deviation of the sampling distribution (also known as the <i>standard error</i>) can take on one of two possible values:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393318.image1.png\" alt=\"image1.png\" width=\"76\" height=\"48\" /></p>\n<p>This is the appropriate choice for a &#8220;small&#8221; sample; for example, the sample size is less than or equal to 5 percent of the population size.</p>\n<p>If the sample is &#8220;large,&#8221; the standard error becomes:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393319.image2.png\" alt=\"image2.png\" width=\"123\" height=\"51\" /></p>\n<p>Probabilities may be computed for the sample mean directly from the standard normal table by applying the following formula:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393320.image3.png\" alt=\"image3.png\" width=\"92\" height=\"51\" /></p>\n"},{"title":"Explore hypothesis testing in business statistics","thumb":null,"image":null,"content":"<p>In statistics, <i>h</i><i>ypothesis testing</i> refers to the process of choosing between competing hypotheses about a probability distribution, based on observed data from the distribution. It&#8217;s a core topic and a fundamental part of the language of statistics.</p>\n<p>Hypothesis testing is a six-step procedure:</p>\n<p>1.    Null hypothesis</p>\n<p>2.    Alternative hypothesis</p>\n<p>3.    Level of significance</p>\n<p>4.    Test statistic</p>\n<p>5.    Critical value(s)</p>\n<p>6.    Decision rule</p>\n<p>The <i>null hypothesis</i> is a statement that&#8217;s assumed to be true unless there&#8217;s strong contradictory evidence. The <i>alternative hypothesis</i> is a statement that will be accepted in place of the null hypothesis if it is rejected.</p>\n<p>The <i>level of significance</i> is chosen to control the probability of a &#8220;Type I&#8221; error; this is the error that results when the null hypothesis is erroneously rejected.</p>\n<p>The <i>test statistic </i>and<i> critical values</i> are used to determine if the null hypothesis should be rejected. The <i>decision rule</i> that is followed is that an &#8220;extreme&#8221; test statistic results in rejection of the null hypothesis. Here, an extreme test statistic is one that lies outside the bounds of the critical value or values.</p>\n<p class=\"Remember\">Hypotheses are often tested about the values of population measures such as the mean and the variance. They are also used to determine if a population follows a specified probability distribution. They also form a major part of regression analysis, where hypotheses are used to validate the results of an estimated regression equation.</p>\n"},{"title":"How businesses use regression analysis statistics","thumb":null,"image":null,"content":"<p>Regression analysis is a statistical tool used for the investigation of relationships between variables. Usually, the investigator seeks to ascertain the causal effect of one variable upon another — the effect of a price increase upon demand, for example, or the effect of changes in the money supply upon the inflation rate.</p>\n<p>Regression analysis is used to estimate the strength and the direction of the relationship between two linearly related variables: X and Y. X is the &#8220;independent&#8221; variable and Y is the &#8220;dependent&#8221; variable.</p>\n<p>The two basic types of regression analysis are:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><b>Simple regression analysis:</b> Used to estimate the relationship between a dependent variable and a single independent variable; for example, the relationship between crop yields and rainfall.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Multiple regression analysis:</b> Used to estimate the relationship between a dependent variable and two or more independent variables; for example, the relationship between the salaries of employees and their experience and education.</p>\n<p class=\"child-para\">Multiple regression analysis introduces several additional complexities but may produce more realistic results than simple regression analysis.</p>\n</li>\n</ul>\n<p>Regression analysis is based on several strong assumptions about the variables that are being estimated. Several key tests are used to ensure that the results are valid, including hypothesis tests. These tests are used to ensure that the regression results are not simply due to random chance but indicate an actual relationship between two or more variables.</p>\n<p>An estimated regression equation may be used for a wide variety of business applications, such as:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Measuring the impact on a corporation&#8217;s profits of an increase in profits</p>\n</li>\n<li>\n<p class=\"first-para\">Understanding how sensitive a corporation&#8217;s sales are to changes in advertising expenditures</p>\n</li>\n<li>\n<p class=\"first-para\">Seeing how a stock price is affected by changes in interest rates</p>\n</li>\n</ul>\n<p>Regression analysis may also be used for forecasting purposes; for example, a regression equation may be used to forecast the future demand for a company&#8217;s products.</p>\n<p>Due to the extreme complexity of regression analysis, it is often implemented through the use of specialized calculators or spreadsheet programs.</p>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-01-31T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":207822},{"headers":{"creationTime":"2018-07-30T20:36:13+00:00","modifiedTime":"2019-06-20T18:42:50+00:00","timestamp":"2022-02-24T17:05:16+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"Important Terms in Game Theory","strippedTitle":"important terms in game theory","slug":"important-terms-game-theory","canonicalUrl":"","seo":{"metaDescription":"","noIndex":0,"noFollow":0},"content":"As with many areas and topics in finite mathematics, there is a very special and specific vocabulary that goes along with game theory. Here are some important and useful terms that you should know.\r\n<ul>\r\n \t<li><b>Payoff matrix:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A matrix whose elements represent all the amounts won or lost by the row player.</span></li>\r\n \t<li><b>Payoff:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> An amount showing as an element in the payoff matrix, which indicates the amount gained or lost by the row player.</span></li>\r\n \t<li><b>Saddle point:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> The element in a payoff matrix that is the smallest in a particular row while, at the same time, the largest in its column. Not all matrices have saddle points.</span></li>\r\n \t<li><b>Strictly determined game:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A game that has a saddle point.</span></li>\r\n \t<li><strong>Strategy:</strong><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A move or moves chosen by a player.</span></li>\r\n \t<li><b>Optimal strategy:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> The strategy that most benefits a player.</span></li>\r\n \t<li><b>Value (expected value) of game:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> The amount representing the result when the best possible strategy is played by each player.</span></li>\r\n \t<li><b>Zero-sum game:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A game where what one player wins, the other loses; no money comes in from the outside or leaves.</span></li>\r\n \t<li><b>Fair game:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A game with a value of 0.</span></li>\r\n \t<li><b>Pure strategy:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A player always chooses the same row or column.</span></li>\r\n \t<li><b>Mixed strategy:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A player changes the choice of row or column with different plays or turns.</span></li>\r\n \t<li><b>Dominated strategy:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A strategy that is never considered because another play is always better. For the row player, a row is dominated by another row if all the corresponding elements are all larger. For the column player, a column is dominated by another column if all the corresponding elements are all smaller.</span></li>\r\n</ul>","description":"As with many areas and topics in finite mathematics, there is a very special and specific vocabulary that goes along with game theory. Here are some important and useful terms that you should know.\r\n<ul>\r\n \t<li><b>Payoff matrix:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A matrix whose elements represent all the amounts won or lost by the row player.</span></li>\r\n \t<li><b>Payoff:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> An amount showing as an element in the payoff matrix, which indicates the amount gained or lost by the row player.</span></li>\r\n \t<li><b>Saddle point:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> The element in a payoff matrix that is the smallest in a particular row while, at the same time, the largest in its column. Not all matrices have saddle points.</span></li>\r\n \t<li><b>Strictly determined game:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A game that has a saddle point.</span></li>\r\n \t<li><strong>Strategy:</strong><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A move or moves chosen by a player.</span></li>\r\n \t<li><b>Optimal strategy:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> The strategy that most benefits a player.</span></li>\r\n \t<li><b>Value (expected value) of game:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> The amount representing the result when the best possible strategy is played by each player.</span></li>\r\n \t<li><b>Zero-sum game:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A game where what one player wins, the other loses; no money comes in from the outside or leaves.</span></li>\r\n \t<li><b>Fair game:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A game with a value of 0.</span></li>\r\n \t<li><b>Pure strategy:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A player always chooses the same row or column.</span></li>\r\n \t<li><b>Mixed strategy:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A player changes the choice of row or column with different plays or turns.</span></li>\r\n \t<li><b>Dominated strategy:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A strategy that is never considered because another play is always better. For the row player, a row is dominated by another row if all the corresponding elements are all larger. For the column player, a column is dominated by another column if all the corresponding elements are all smaller.</span></li>\r\n</ul>","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":34229,"title":"Calculation & Analysis","slug":"calculation-analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}],"fromCategory":[{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282200,"slug":"finite-math-for-dummies","isbn":"9781119476368","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"amazon":{"default":"https://www.amazon.com/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119476364-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/finite-math-for-dummies-cover-9781119476368-203x255.jpg","width":203,"height":255},"title":"Finite Math For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <p><b data-author-id=\"8985\">Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-6217bacc418ce\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-6217bacc4240a\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":254831},{"headers":{"creationTime":"2018-07-30T20:31:43+00:00","modifiedTime":"2018-07-30T20:31:43+00:00","timestamp":"2022-02-24T17:04:40+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"How to Create a Matrix from a Transition Diagram","strippedTitle":"how to create a matrix from a transition diagram","slug":"create-matrix-transition-diagram","canonicalUrl":"","seo":{"metaDescription":"","noIndex":0,"noFollow":0},"content":"When you encounter a matrix problem in finite math, a nice way to illustrate the transition from one state to another is to use a transition diagram. The different states are represented by circles, and the probability of going from one state to another is shown by using curves with arrows.\r\n\r\nThe transition diagram in the following figure shows how an insurance company classifies its drivers: no accidents, one accident, or two or more accidents. This information could help the company determine the insurance premium rates.\r\n\r\n[caption id=\"attachment_254823\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254823\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1402.jpg\" alt=\"FNTMath-transition-diagram\" width=\"515\" height=\"363\" /> Eighty percent of the no-accident drivers probably won’t have an accident.[/caption]\r\n\r\nYou see that 80% of the drivers who haven’t had an accident aren’t expected to have an accident the next year. Fifteen percent of those drivers have one accident, and 5% have two or more accidents. Seventy percent of those who have had one accident aren’t expected to have an accident the next year but have to stay in the one-accident classification. And those in the two-or-more accident class have to stay there.\r\n\r\nTo create a transition matrix representing the drivers, use the percentages to show going from one state to another.\r\n\r\n<img class=\"alignnone size-full wp-image-254726\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2401.jpg\" alt=\"FNTMATH_2401\" width=\"264\" height=\"100\" />\r\n\r\nWhat is the long-term expectation for these drivers? First, let the transition matrix be D.\r\n\r\n<img class=\"alignnone size-full wp-image-254727\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2402.jpg\" alt=\"FNTMATH_2402\" width=\"155\" height=\"68\" />\r\n\r\nThen, some of the powers of D are\r\n\r\n<img class=\"alignnone size-full wp-image-254728\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2403.jpg\" alt=\"FNTMATH_2403\" width=\"253\" height=\"297\" />\r\n\r\nAt the end of ten years, using the drivers in the initial study, you have\r\n\r\n<img class=\"alignnone size-full wp-image-254729\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2404.jpg\" alt=\"FNTMATH_2404\" width=\"267\" height=\"100\" />\r\n\r\nWhat this tells the insurance company is that, in ten years, about 11% of the original no-accident drivers will still not have had an accident. Only 3% of the one-accident drivers will still have had only that one accident. This situation doesn’t allow for the drivers to move back or earn forgiveness; a one-accident driver can’t be a no-accident driver using this model. Of course, different insurance agencies have different policies, putting drivers in better standing after a set number of accident-free years. And new policyholders are added to make this picture rosier. This just shows the pattern for a particular set of drivers after a certain number of years.","description":"When you encounter a matrix problem in finite math, a nice way to illustrate the transition from one state to another is to use a transition diagram. The different states are represented by circles, and the probability of going from one state to another is shown by using curves with arrows.\r\n\r\nThe transition diagram in the following figure shows how an insurance company classifies its drivers: no accidents, one accident, or two or more accidents. This information could help the company determine the insurance premium rates.\r\n\r\n[caption id=\"attachment_254823\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254823\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1402.jpg\" alt=\"FNTMath-transition-diagram\" width=\"515\" height=\"363\" /> Eighty percent of the no-accident drivers probably won’t have an accident.[/caption]\r\n\r\nYou see that 80% of the drivers who haven’t had an accident aren’t expected to have an accident the next year. Fifteen percent of those drivers have one accident, and 5% have two or more accidents. Seventy percent of those who have had one accident aren’t expected to have an accident the next year but have to stay in the one-accident classification. And those in the two-or-more accident class have to stay there.\r\n\r\nTo create a transition matrix representing the drivers, use the percentages to show going from one state to another.\r\n\r\n<img class=\"alignnone size-full wp-image-254726\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2401.jpg\" alt=\"FNTMATH_2401\" width=\"264\" height=\"100\" />\r\n\r\nWhat is the long-term expectation for these drivers? First, let the transition matrix be D.\r\n\r\n<img class=\"alignnone size-full wp-image-254727\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2402.jpg\" alt=\"FNTMATH_2402\" width=\"155\" height=\"68\" />\r\n\r\nThen, some of the powers of D are\r\n\r\n<img class=\"alignnone size-full wp-image-254728\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2403.jpg\" alt=\"FNTMATH_2403\" width=\"253\" height=\"297\" />\r\n\r\nAt the end of ten years, using the drivers in the initial study, you have\r\n\r\n<img class=\"alignnone size-full wp-image-254729\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2404.jpg\" alt=\"FNTMATH_2404\" width=\"267\" height=\"100\" />\r\n\r\nWhat this tells the insurance company is that, in ten years, about 11% of the original no-accident drivers will still not have had an accident. Only 3% of the one-accident drivers will still have had only that one accident. This situation doesn’t allow for the drivers to move back or earn forgiveness; a one-accident driver can’t be a no-accident driver using this model. Of course, different insurance agencies have different policies, putting drivers in better standing after a set number of accident-free years. And new policyholders are added to make this picture rosier. This just shows the pattern for a particular set of drivers after a certain number of years.","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":34229,"title":"Calculation & Analysis","slug":"calculation-analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}],"fromCategory":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282200,"slug":"finite-math-for-dummies","isbn":"9781119476368","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"amazon":{"default":"https://www.amazon.com/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119476364-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/finite-math-for-dummies-cover-9781119476368-203x255.jpg","width":203,"height":255},"title":"Finite Math For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <p><b data-author-id=\"8985\">Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-6217baa89bdfd\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-6217baa89c799\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":254827},{"headers":{"creationTime":"2018-07-30T20:26:00+00:00","modifiedTime":"2018-07-30T20:26:00+00:00","timestamp":"2022-02-24T17:04:40+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"How to Use Transition Matrices","strippedTitle":"how to use transition matrices","slug":"use-transition-matrices","canonicalUrl":"","seo":{"metaDescription":"","noIndex":0,"noFollow":0},"content":"If your finite math instructor asks you to predict the likelihood of an action repeating over time, you may need to use a transition matrix to do this. A <em>transition matrix</em> consists of a square matrix that gives the probabilities of different states going from one to another.\r\n\r\nWith a transition matrix, you can perform matrix multiplication and determine trends, if there are any, and make predications.\r\n\r\nConsider the table showing the purchasing patterns involving different cereals. You see all the percentages showing the probability of going from one state to another, but which of the cereals does the consumer actually end up buying most frequently in the long run?\r\n\r\n<img class=\"alignnone size-full wp-image-254717\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2301.jpg\" alt=\"FNTMATH_2301\" width=\"363\" height=\"103\" />\r\n\r\nOne way to look at continued purchasing is to create a tree diagram. In the following figure, you see two consecutive “rounds” of purchases.\r\n\r\n[caption id=\"attachment_254822\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254822\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1401.jpg\" alt=\"FNTMath-transition-matrices\" width=\"515\" height=\"320\" /> Which kind of cereal will the consumer buy?[/caption]\r\n\r\nIf you want the probability that the consumer purchases Kicks first, tries it again or something else, and then purchases Kicks the next time, add up the , , and branches: , or 38% of the time. If you want the probability that the consumer purchases Cheery A’s first, tries something else or repeats Cheery A’s, and then tries Corn Flecks, add up the , , and branches. This comes out to , or almost 26% of the time.\r\n\r\nThe tree is helpful in that it shows you what the choices are and how the percentages work in determining patterns, but there’s a much easier and neater way to compute these values.\r\n\r\nTo perform computations and study this further, create a transition matrix, referring back to the chart showing purchases and using the decimal values of the percentages. Name it matrix C.\r\n\r\n<img class=\"alignnone size-full wp-image-254718\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2302.jpg\" alt=\"FNTMATH_2302\" width=\"155\" height=\"68\" />\r\n\r\nNext, use matrix multiplication to find <em>C</em>². As a quick hint, when multiplying matrices, you find the element in the first row, first column of the product, labeled <em>c</em>11, when you multiply the elements in the first row of the first matrix times the corresponding elements in the first column of the second matrix and then add up the products.\r\n<p class=\"article-tips remember\">In a matrix A, the element in the <em>n</em>th row, <em>k</em>th column is labeled <em>ank</em>.</p>\r\nThe element in the first row and second column of the product, <em>c</em>12, uses the elements in the first row of the first matrix and second column of the second matrix, and so on for the rest of the elements.\r\n\r\n<img class=\"alignnone size-full wp-image-254719\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2303.jpg\" alt=\"FNTMATH_2303\" width=\"316\" height=\"68\" />\r\n\r\nSo, you take the first row of the left matrix times the first column of the second matrix to get\r\n\r\n<img class=\"alignnone size-full wp-image-254720\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2304.jpg\" alt=\"FNTMATH_2304\" width=\"207\" height=\"40\" />\r\n\r\nYes. This is the same computation as was done using the tree to find the probability that a consumer starting with Kicks would return to it in two more purchases.\r\n\r\nPerforming the matrix multiplication, you have\r\n\r\n<img class=\"alignnone size-full wp-image-254721\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2305.jpg\" alt=\"FNTMATH_2305\" width=\"191\" height=\"68\" />\r\n\r\nContinuing this multiplication process, by the time <em>C</em>6 appears (the chances of buying a particular cereal at the fifth purchase time after the initial purchase), a pattern emerges.\r\n\r\n<img class=\"alignnone size-full wp-image-254722\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2306.jpg\" alt=\"FNTMATH_2306\" width=\"249\" height=\"68\" />\r\n\r\nNotice that the numbers in each column round to the same three decimal places. This is going to become even clearer, using higher powers of C, until some nth matrix power becomes\r\n\r\n<img class=\"alignnone size-full wp-image-254723\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2307.jpg\" alt=\"FNTMATH_2307\" width=\"184\" height=\"68\" />\r\n\r\nThe matrix shows you the pattern or trend.\r\n\r\n<img class=\"alignnone size-full wp-image-254724\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2308.jpg\" alt=\"FNTMATH_2308\" width=\"239\" height=\"100\" />\r\n\r\nNo matter which cereal the consumer bought first, in the long run there’s a 35.3% chance that she’ll purchase Kicks, a 38.4% chance that she’ll purchase Cheery A’s, and a 26.3% chance that she’ll purchase Corn Flecks. This transition matrix has reached an equilibrium, where it won’t change with more repeated multiplication. You can write this situation with a single-line matrix:\r\n\r\n<img class=\"alignnone size-full wp-image-254725\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2309.jpg\" alt=\"FNTMATH_2309\" width=\"153\" height=\"65\" />","description":"If your finite math instructor asks you to predict the likelihood of an action repeating over time, you may need to use a transition matrix to do this. A <em>transition matrix</em> consists of a square matrix that gives the probabilities of different states going from one to another.\r\n\r\nWith a transition matrix, you can perform matrix multiplication and determine trends, if there are any, and make predications.\r\n\r\nConsider the table showing the purchasing patterns involving different cereals. You see all the percentages showing the probability of going from one state to another, but which of the cereals does the consumer actually end up buying most frequently in the long run?\r\n\r\n<img class=\"alignnone size-full wp-image-254717\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2301.jpg\" alt=\"FNTMATH_2301\" width=\"363\" height=\"103\" />\r\n\r\nOne way to look at continued purchasing is to create a tree diagram. In the following figure, you see two consecutive “rounds” of purchases.\r\n\r\n[caption id=\"attachment_254822\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254822\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1401.jpg\" alt=\"FNTMath-transition-matrices\" width=\"515\" height=\"320\" /> Which kind of cereal will the consumer buy?[/caption]\r\n\r\nIf you want the probability that the consumer purchases Kicks first, tries it again or something else, and then purchases Kicks the next time, add up the , , and branches: , or 38% of the time. If you want the probability that the consumer purchases Cheery A’s first, tries something else or repeats Cheery A’s, and then tries Corn Flecks, add up the , , and branches. This comes out to , or almost 26% of the time.\r\n\r\nThe tree is helpful in that it shows you what the choices are and how the percentages work in determining patterns, but there’s a much easier and neater way to compute these values.\r\n\r\nTo perform computations and study this further, create a transition matrix, referring back to the chart showing purchases and using the decimal values of the percentages. Name it matrix C.\r\n\r\n<img class=\"alignnone size-full wp-image-254718\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2302.jpg\" alt=\"FNTMATH_2302\" width=\"155\" height=\"68\" />\r\n\r\nNext, use matrix multiplication to find <em>C</em>². As a quick hint, when multiplying matrices, you find the element in the first row, first column of the product, labeled <em>c</em>11, when you multiply the elements in the first row of the first matrix times the corresponding elements in the first column of the second matrix and then add up the products.\r\n<p class=\"article-tips remember\">In a matrix A, the element in the <em>n</em>th row, <em>k</em>th column is labeled <em>ank</em>.</p>\r\nThe element in the first row and second column of the product, <em>c</em>12, uses the elements in the first row of the first matrix and second column of the second matrix, and so on for the rest of the elements.\r\n\r\n<img class=\"alignnone size-full wp-image-254719\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2303.jpg\" alt=\"FNTMATH_2303\" width=\"316\" height=\"68\" />\r\n\r\nSo, you take the first row of the left matrix times the first column of the second matrix to get\r\n\r\n<img class=\"alignnone size-full wp-image-254720\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2304.jpg\" alt=\"FNTMATH_2304\" width=\"207\" height=\"40\" />\r\n\r\nYes. This is the same computation as was done using the tree to find the probability that a consumer starting with Kicks would return to it in two more purchases.\r\n\r\nPerforming the matrix multiplication, you have\r\n\r\n<img class=\"alignnone size-full wp-image-254721\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2305.jpg\" alt=\"FNTMATH_2305\" width=\"191\" height=\"68\" />\r\n\r\nContinuing this multiplication process, by the time <em>C</em>6 appears (the chances of buying a particular cereal at the fifth purchase time after the initial purchase), a pattern emerges.\r\n\r\n<img class=\"alignnone size-full wp-image-254722\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2306.jpg\" alt=\"FNTMATH_2306\" width=\"249\" height=\"68\" />\r\n\r\nNotice that the numbers in each column round to the same three decimal places. This is going to become even clearer, using higher powers of C, until some nth matrix power becomes\r\n\r\n<img class=\"alignnone size-full wp-image-254723\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2307.jpg\" alt=\"FNTMATH_2307\" width=\"184\" height=\"68\" />\r\n\r\nThe matrix shows you the pattern or trend.\r\n\r\n<img class=\"alignnone size-full wp-image-254724\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2308.jpg\" alt=\"FNTMATH_2308\" width=\"239\" height=\"100\" />\r\n\r\nNo matter which cereal the consumer bought first, in the long run there’s a 35.3% chance that she’ll purchase Kicks, a 38.4% chance that she’ll purchase Cheery A’s, and a 26.3% chance that she’ll purchase Corn Flecks. This transition matrix has reached an equilibrium, where it won’t change with more repeated multiplication. You can write this situation with a single-line matrix:\r\n\r\n<img class=\"alignnone size-full wp-image-254725\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2309.jpg\" alt=\"FNTMATH_2309\" width=\"153\" height=\"65\" />","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":34229,"title":"Calculation & Analysis","slug":"calculation-analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}],"fromCategory":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282200,"slug":"finite-math-for-dummies","isbn":"9781119476368","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"amazon":{"default":"https://www.amazon.com/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119476364-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/finite-math-for-dummies-cover-9781119476368-203x255.jpg","width":203,"height":255},"title":"Finite Math For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <p><b data-author-id=\"8985\">Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-6217baa893b96\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-6217baa894533\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":254821},{"headers":{"creationTime":"2018-07-30T20:13:00+00:00","modifiedTime":"2018-07-30T20:13:00+00:00","timestamp":"2022-02-24T17:04:40+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"How to Analyze Arguments with Euler Diagrams","strippedTitle":"how to analyze arguments with euler diagrams","slug":"analyze-arguments-euler-diagrams","canonicalUrl":"","seo":{"metaDescription":"","noIndex":0,"noFollow":0},"content":"On a finite math exam, you may be asked to analyze an argument with a visual approach using an Euler diagram. This pictorial technique is used to check to see whether an argument is valid.\r\n<p class=\"article-tips remember\">An argument can be classified as either valid or invalid. A valid argument occurs in situations where if the premises are true, then the conclusion must also be true. And an argument can be valid even if the conclusion is false.</p>\r\nThe following argument has two premises: (1) “All dogs have fleas.” (2) “Hank is a dog.” The conclusion is that, therefore, Hank has fleas.\r\n\r\nThese arguments usually have the following format with the premises listed first and the conclusion under a horizontal line:\r\n\r\n<img class=\"alignnone size-full wp-image-254714\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2201.jpg\" alt=\"FNTMATH_2201\" width=\"199\" height=\"79\" />\r\n\r\nUsing an Euler diagram to analyze this argument, draw a circle to contain all objects that have fleas. Inside the circle, put another circle to contain all dogs. And inside the circle of dogs, put Hank. The figure illustrates this approach.\r\n\r\n[caption id=\"attachment_254815\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254815\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1301.jpg\" alt=\"1119476364-fg1301\" width=\"515\" height=\"543\" /> Poor Hank has fleas.[/caption]\r\n\r\nThe argument isn’t necessarily true, because you know that not all dogs have fleas. All this shows is that the argument is <em>valid</em>. If the two premises are true, then the conclusion must be true.\r\n\r\nNow consider an argument involving rectangles and triangles. A polygon is a figure made up of line segments connected at their endpoints.\r\n\r\n<img class=\"alignnone size-full wp-image-254715\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2202.jpg\" alt=\"FNTMATH_2202\" width=\"172\" height=\"81\" />\r\n\r\nWhen analyzing the validity of this argument, the Euler diagram starts with a circle containing all polygons, as shown here.\r\n\r\n[caption id=\"attachment_254816\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254816\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1302.jpg\" alt=\"1119476364-fg1302\" width=\"515\" height=\"543\" /> Two types of polygons.[/caption]\r\n\r\nTwo circles are drawn inside the larger circle—one containing rectangles and the other triangles. The two circles don’t overlap, because rectangles have four sides, and triangles have three sides.\r\n\r\nThe argument is <em>invalid</em>. Rectangles are not triangles—not even sometimes.\r\n\r\nArguments can have more than two premises. For example:\r\n\r\n<img class=\"alignnone size-full wp-image-254716\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2203.jpg\" alt=\"FNTMATH_2203\" width=\"307\" height=\"112\" />\r\n\r\nOne Euler diagram that can represent this situation has three intersecting circles, as shown here.\r\n\r\n[caption id=\"attachment_254817\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254817\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1303.jpg\" alt=\"1119476364-fg1303\" width=\"515\" height=\"558\" /> President Abraham Lincoln and other Illinois lawyers.[/caption]\r\n\r\nAs you can see from the diagram, there can be presidents born in Kentucky who were not lawyers in Illinois and there can be presidents who were lawyers in Illinois but not born in Kentucky. The argument is <em>invalid</em>. To be valid, it must always be true.","description":"On a finite math exam, you may be asked to analyze an argument with a visual approach using an Euler diagram. This pictorial technique is used to check to see whether an argument is valid.\r\n<p class=\"article-tips remember\">An argument can be classified as either valid or invalid. A valid argument occurs in situations where if the premises are true, then the conclusion must also be true. And an argument can be valid even if the conclusion is false.</p>\r\nThe following argument has two premises: (1) “All dogs have fleas.” (2) “Hank is a dog.” The conclusion is that, therefore, Hank has fleas.\r\n\r\nThese arguments usually have the following format with the premises listed first and the conclusion under a horizontal line:\r\n\r\n<img class=\"alignnone size-full wp-image-254714\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2201.jpg\" alt=\"FNTMATH_2201\" width=\"199\" height=\"79\" />\r\n\r\nUsing an Euler diagram to analyze this argument, draw a circle to contain all objects that have fleas. Inside the circle, put another circle to contain all dogs. And inside the circle of dogs, put Hank. The figure illustrates this approach.\r\n\r\n[caption id=\"attachment_254815\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254815\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1301.jpg\" alt=\"1119476364-fg1301\" width=\"515\" height=\"543\" /> Poor Hank has fleas.[/caption]\r\n\r\nThe argument isn’t necessarily true, because you know that not all dogs have fleas. All this shows is that the argument is <em>valid</em>. If the two premises are true, then the conclusion must be true.\r\n\r\nNow consider an argument involving rectangles and triangles. A polygon is a figure made up of line segments connected at their endpoints.\r\n\r\n<img class=\"alignnone size-full wp-image-254715\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2202.jpg\" alt=\"FNTMATH_2202\" width=\"172\" height=\"81\" />\r\n\r\nWhen analyzing the validity of this argument, the Euler diagram starts with a circle containing all polygons, as shown here.\r\n\r\n[caption id=\"attachment_254816\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254816\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1302.jpg\" alt=\"1119476364-fg1302\" width=\"515\" height=\"543\" /> Two types of polygons.[/caption]\r\n\r\nTwo circles are drawn inside the larger circle—one containing rectangles and the other triangles. The two circles don’t overlap, because rectangles have four sides, and triangles have three sides.\r\n\r\nThe argument is <em>invalid</em>. Rectangles are not triangles—not even sometimes.\r\n\r\nArguments can have more than two premises. For example:\r\n\r\n<img class=\"alignnone size-full wp-image-254716\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2203.jpg\" alt=\"FNTMATH_2203\" width=\"307\" height=\"112\" />\r\n\r\nOne Euler diagram that can represent this situation has three intersecting circles, as shown here.\r\n\r\n[caption id=\"attachment_254817\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254817\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1303.jpg\" alt=\"1119476364-fg1303\" width=\"515\" height=\"558\" /> President Abraham Lincoln and other Illinois lawyers.[/caption]\r\n\r\nAs you can see from the diagram, there can be presidents born in Kentucky who were not lawyers in Illinois and there can be presidents who were lawyers in Illinois but not born in Kentucky. The argument is <em>invalid</em>. To be valid, it must always be true.","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":34229,"title":"Calculation & Analysis","slug":"calculation-analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}],"fromCategory":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282200,"slug":"finite-math-for-dummies","isbn":"9781119476368","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"amazon":{"default":"https://www.amazon.com/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119476364-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/finite-math-for-dummies-cover-9781119476368-203x255.jpg","width":203,"height":255},"title":"Finite Math For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <p><b data-author-id=\"8985\">Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-6217baa88b21e\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-6217baa88bbb3\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":254814},{"headers":{"creationTime":"2018-07-30T20:04:45+00:00","modifiedTime":"2018-07-30T20:04:45+00:00","timestamp":"2022-02-24T17:04:40+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"How to Analyze Compound Statements","strippedTitle":"how to analyze compound statements","slug":"analyze-compound-statements","canonicalUrl":"","seo":{"metaDescription":"","noIndex":0,"noFollow":0},"content":"If your finite math instructor asks you to analyze a compound statement, you can try using a truth table to do this. Not every topic in a discussion can be turned into a compound statement and analyzed for its truth that way, but using logic and truth values is a good technique to use when possible.\r\n\r\nConsider the compound statement\r\n\r\n<img class=\"alignnone size-full wp-image-254706\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2101.jpg\" alt=\"FNTMATH_2101\" width=\"84\" height=\"25\" />\r\n\r\nWhen constructing a truth table, you start with the basic p and q columns. Then you add a ~ q column followed by a column\r\n\r\n<img class=\"alignnone size-full wp-image-254707\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2102.jpg\" alt=\"FNTMATH_2102\" width=\"45\" height=\"17\" />\r\n\r\nBefore you can perform the conjunction, ^, you need a ~ p column. Here’s a step-by-step procedure.\r\n<ol>\r\n \t<li><strong>Start with a basic <em>p</em> and<em> q</em> and then add ~ <em>q</em>.</strong>\r\n<img class=\"alignnone size-full wp-image-254708\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2103.jpg\" alt=\"FNTMATH_2103\" width=\"132\" height=\"100\" /></li>\r\n \t<li>When adding the\r\n<img class=\"alignnone size-full wp-image-254709\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2104.jpg\" alt=\"FNTMATH_2104\" width=\"45\" height=\"17\" />\r\ncolumn, perform the disjunction on the first and third columns.\r\nRemember, with disjunctions, the statement is false only when both component statements are false.\r\n<img class=\"alignnone size-full wp-image-254710\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2105.jpg\" alt=\"FNTMATH_2105\" width=\"131\" height=\"120\" /></li>\r\n \t<li><strong>Add the ~ p column.</strong>\r\n<img class=\"alignnone size-full wp-image-254711\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2106.jpg\" alt=\"FNTMATH_2106\" width=\"161\" height=\"100\" /></li>\r\n \t<li>Add the\r\n<img class=\"alignnone size-full wp-image-254712\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2107.jpg\" alt=\"FNTMATH_2107\" width=\"85\" height=\"25\" />\r\ncolumn, which shows the conjunction of the fourth and fifth columns.\r\n<img class=\"alignnone size-full wp-image-254713\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2108.jpg\" alt=\"FNTMATH_2108\" width=\"255\" height=\"124\" /></li>\r\n</ol>\r\n<p class=\"article-tips remember\">The conjunction is true only when the two component statements are true. This complex statement is only true when both original statements are false.</p>","description":"If your finite math instructor asks you to analyze a compound statement, you can try using a truth table to do this. Not every topic in a discussion can be turned into a compound statement and analyzed for its truth that way, but using logic and truth values is a good technique to use when possible.\r\n\r\nConsider the compound statement\r\n\r\n<img class=\"alignnone size-full wp-image-254706\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2101.jpg\" alt=\"FNTMATH_2101\" width=\"84\" height=\"25\" />\r\n\r\nWhen constructing a truth table, you start with the basic p and q columns. Then you add a ~ q column followed by a column\r\n\r\n<img class=\"alignnone size-full wp-image-254707\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2102.jpg\" alt=\"FNTMATH_2102\" width=\"45\" height=\"17\" />\r\n\r\nBefore you can perform the conjunction, ^, you need a ~ p column. Here’s a step-by-step procedure.\r\n<ol>\r\n \t<li><strong>Start with a basic <em>p</em> and<em> q</em> and then add ~ <em>q</em>.</strong>\r\n<img class=\"alignnone size-full wp-image-254708\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2103.jpg\" alt=\"FNTMATH_2103\" width=\"132\" height=\"100\" /></li>\r\n \t<li>When adding the\r\n<img class=\"alignnone size-full wp-image-254709\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2104.jpg\" alt=\"FNTMATH_2104\" width=\"45\" height=\"17\" />\r\ncolumn, perform the disjunction on the first and third columns.\r\nRemember, with disjunctions, the statement is false only when both component statements are false.\r\n<img class=\"alignnone size-full wp-image-254710\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2105.jpg\" alt=\"FNTMATH_2105\" width=\"131\" height=\"120\" /></li>\r\n \t<li><strong>Add the ~ p column.</strong>\r\n<img class=\"alignnone size-full wp-image-254711\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2106.jpg\" alt=\"FNTMATH_2106\" width=\"161\" height=\"100\" /></li>\r\n \t<li>Add the\r\n<img class=\"alignnone size-full wp-image-254712\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2107.jpg\" alt=\"FNTMATH_2107\" width=\"85\" height=\"25\" />\r\ncolumn, which shows the conjunction of the fourth and fifth columns.\r\n<img class=\"alignnone size-full wp-image-254713\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2108.jpg\" alt=\"FNTMATH_2108\" width=\"255\" height=\"124\" /></li>\r\n</ol>\r\n<p class=\"article-tips remember\">The conjunction is true only when the two component statements are true. This complex statement is only true when both original statements are false.</p>","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":34229,"title":"Calculation & Analysis","slug":"calculation-analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}],"fromCategory":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282200,"slug":"finite-math-for-dummies","isbn":"9781119476368","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"amazon":{"default":"https://www.amazon.com/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119476364-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/finite-math-for-dummies-cover-9781119476368-203x255.jpg","width":203,"height":255},"title":"Finite Math For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <p><b data-author-id=\"8985\">Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-6217baa882c92\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-6217baa8835fc\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":254811},{"headers":{"creationTime":"2018-07-30T19:56:55+00:00","modifiedTime":"2018-07-30T19:56:55+00:00","timestamp":"2022-02-24T17:04:40+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"How to Calculate Monthly Payments for a Sinking Fund","strippedTitle":"how to calculate monthly payments for a sinking fund","slug":"calculate-monthly-payments-sinking-fund","canonicalUrl":"","seo":{"metaDescription":"","noIndex":0,"noFollow":0},"content":"A big part of finite math involves working through financial problems. Some of these problems may seem complex—like calculating the monthly deposits required to maintain a sinking fund. Fortunately, there’s a special formula you can use to find the answer.\r\n\r\nA <em>sinking fund</em> is usually used to accumulate money to fund a future expense or a way to retire a debt. You can use a sinking fund to pay off a loan in one lump sum at the end of a set amount of time while making just interest payments in the meantime.\r\n\r\nFor example, a friend borrows $10,000 to purchase a boat and agrees to pay the full amount back in one payment, ten years from now. In the meantime, he agrees to pay interest monthly on the $10,000 at an annual rate of 12%. He also sets up a sinking fund to accumulate the lump-sum payment. The sinking fund earns 9% interest, compounded monthly. How much does he pay monthly? The monthly amount is both the interest to the lender and a deposit into the sinking fund.\r\n\r\nThe interest to the lender is based on an annual rate of 12%. Using the simple interest formula, <em>I</em> = <em>Prt</em>, you have <em>I</em> = 10,000(0.12)(1) = 1,200 per year. Because he plans to make monthly payments, you divide by 12 so $100 per month goes for the interest payments.\r\n\r\nNext, you compute the amount to be deposited in the sinking fund each month.\r\n<p class=\"article-tips remember\">The formula for a sinking fund payment is</p>\r\n<img class=\"alignnone size-full wp-image-254704\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2001.jpg\" alt=\"FNTMATH_2001\" width=\"84\" height=\"44\" />\r\n\r\nwhere <em>P</em> is the amount of the payment, <em>A</em> is the amount to be accumulated, <em>i</em> is the interest rate per time period, and<em> n</em> is the number of time periods.\r\n\r\nUsing the formula to determine the monthly payment into the sinking fund, the amount, A, is $10,000, and the interest per pay period is 9% divided by 12, because it’s compounded monthly. The number of time periods over the ten years is 120\r\n\r\n<img class=\"alignnone size-full wp-image-254705\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2002.jpg\" alt=\"FNTMATH_2002\" width=\"419\" height=\"44\" />\r\n\r\nSo, the monthly payment into the sinking fund is about $51.68. Add that to the interest payments, and the monthly commitment is $151.68. In ten years, the monthly payments will end.","description":"A big part of finite math involves working through financial problems. Some of these problems may seem complex—like calculating the monthly deposits required to maintain a sinking fund. Fortunately, there’s a special formula you can use to find the answer.\r\n\r\nA <em>sinking fund</em> is usually used to accumulate money to fund a future expense or a way to retire a debt. You can use a sinking fund to pay off a loan in one lump sum at the end of a set amount of time while making just interest payments in the meantime.\r\n\r\nFor example, a friend borrows $10,000 to purchase a boat and agrees to pay the full amount back in one payment, ten years from now. In the meantime, he agrees to pay interest monthly on the $10,000 at an annual rate of 12%. He also sets up a sinking fund to accumulate the lump-sum payment. The sinking fund earns 9% interest, compounded monthly. How much does he pay monthly? The monthly amount is both the interest to the lender and a deposit into the sinking fund.\r\n\r\nThe interest to the lender is based on an annual rate of 12%. Using the simple interest formula, <em>I</em> = <em>Prt</em>, you have <em>I</em> = 10,000(0.12)(1) = 1,200 per year. Because he plans to make monthly payments, you divide by 12 so $100 per month goes for the interest payments.\r\n\r\nNext, you compute the amount to be deposited in the sinking fund each month.\r\n<p class=\"article-tips remember\">The formula for a sinking fund payment is</p>\r\n<img class=\"alignnone size-full wp-image-254704\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2001.jpg\" alt=\"FNTMATH_2001\" width=\"84\" height=\"44\" />\r\n\r\nwhere <em>P</em> is the amount of the payment, <em>A</em> is the amount to be accumulated, <em>i</em> is the interest rate per time period, and<em> n</em> is the number of time periods.\r\n\r\nUsing the formula to determine the monthly payment into the sinking fund, the amount, A, is $10,000, and the interest per pay period is 9% divided by 12, because it’s compounded monthly. The number of time periods over the ten years is 120\r\n\r\n<img class=\"alignnone size-full wp-image-254705\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2002.jpg\" alt=\"FNTMATH_2002\" width=\"419\" height=\"44\" />\r\n\r\nSo, the monthly payment into the sinking fund is about $51.68. Add that to the interest payments, and the monthly commitment is $151.68. In ten years, the monthly payments will end.","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":34229,"title":"Calculation & Analysis","slug":"calculation-analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}}],"fromCategory":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282200,"slug":"finite-math-for-dummies","isbn":"9781119476368","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"amazon":{"default":"https://www.amazon.com/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119476364-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/finite-math-for-dummies-cover-9781119476368-203x255.jpg","width":203,"height":255},"title":"Finite Math For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <p><b data-author-id=\"8985\">Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-6217baa87a9d2\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-6217baa87b3bf\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":254808},{"headers":{"creationTime":"2018-07-30T19:50:44+00:00","modifiedTime":"2018-07-30T19:50:44+00:00","timestamp":"2022-02-24T17:04:40+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"How to Calculate the Starting Value of an Annuity","strippedTitle":"how to calculate the starting value of an annuity","slug":"calculate-starting-value-annuity","canonicalUrl":"","seo":{"metaDescription":"","noIndex":0,"noFollow":0},"content":"In a finite math course, you’ll often be asked to use mathematical formulas to solve real-world problems. A good example of this is calculating the starting value of an annuity.\r\n\r\nSay that you’re planning an around-the-world-trip, and your big concern is how to fund this adventure. One way is to have a fund from which you can withdraw a certain amount of money at regular intervals and have just enough money when you’re finished, where the fund goes down to zero.\r\n\r\nYou’re going to sail around the world on your 40-foot sloop. You’re estimating it will take seven years, with all the visiting, sightseeing, and other activities. You need to set up an annuity from which you can withdraw monthly amounts to help with the expenses. You want to have $2,000 available each month and have the balance be zero at the end of the seven years. How much should you put in your annuity account?\r\n<p class=\"article-tips remember\">The present value of an annuity is determined with</p>\r\n<img class=\"alignnone size-full wp-image-254701\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_1901.jpg\" alt=\"FNTMATH_1901\" width=\"116\" height=\"49\" />\r\n\r\nwhere <em>V</em> is the value or amount needed to be deposited into the account, <em>P</em> is the payment or amount withdrawn periodically, <em>i</em> is the interest each time period, and <em>m</em> is the number of time periods.\r\n\r\nYou find a broker who can get you 9% interest on your deposit. You want to withdraw monthly, so that will be 7 times 12, or 84 payments or time periods. The interest rate each time period is found with <em>r/n</em>, which is 9% divided by 12, or 0.0075%. Using the formula for the present value, that comes to\r\n\r\n<img class=\"alignnone size-full wp-image-254702\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_1902.jpg\" alt=\"FNTMATH_1902\" width=\"337\" height=\"95\" />\r\n\r\nYou need to deposit more than $124,000 to be able to make your regular withdrawals and have a balance of zero at the end of seven years. What is $2,000 per month for seven years, if you aren’t withdrawing from an annuity? Multiply\r\n\r\n<img class=\"alignnone size-full wp-image-254703\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_1903.jpg\" alt=\"FNTMATH_1903\" width=\"157\" height=\"19\" />\r\n\r\nSounds like a good deal—if you can come up with that initial deposit.","description":"In a finite math course, you’ll often be asked to use mathematical formulas to solve real-world problems. A good example of this is calculating the starting value of an annuity.\r\n\r\nSay that you’re planning an around-the-world-trip, and your big concern is how to fund this adventure. One way is to have a fund from which you can withdraw a certain amount of money at regular intervals and have just enough money when you’re finished, where the fund goes down to zero.\r\n\r\nYou’re going to sail around the world on your 40-foot sloop. You’re estimating it will take seven years, with all the visiting, sightseeing, and other activities. You need to set up an annuity from which you can withdraw monthly amounts to help with the expenses. You want to have $2,000 available each month and have the balance be zero at the end of the seven years. How much should you put in your annuity account?\r\n<p class=\"article-tips remember\">The present value of an annuity is determined with</p>\r\n<img class=\"alignnone size-full wp-image-254701\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_1901.jpg\" alt=\"FNTMATH_1901\" width=\"116\" height=\"49\" />\r\n\r\nwhere <em>V</em> is the value or amount needed to be deposited into the account, <em>P</em> is the payment or amount withdrawn periodically, <em>i</em> is the interest each time period, and <em>m</em> is the number of time periods.\r\n\r\nYou find a broker who can get you 9% interest on your deposit. You want to withdraw monthly, so that will be 7 times 12, or 84 payments or time periods. The interest rate each time period is found with <em>r/n</em>, which is 9% divided by 12, or 0.0075%. Using the formula for the present value, that comes to\r\n\r\n<img class=\"alignnone size-full wp-image-254702\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_1902.jpg\" alt=\"FNTMATH_1902\" width=\"337\" height=\"95\" />\r\n\r\nYou need to deposit more than $124,000 to be able to make your regular withdrawals and have a balance of zero at the end of seven years. What is $2,000 per month for seven years, if you aren’t withdrawing from an annuity? Multiply\r\n\r\n<img class=\"alignnone size-full wp-image-254703\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_1903.jpg\" alt=\"FNTMATH_1903\" width=\"157\" height=\"19\" />\r\n\r\nSounds like a good deal—if you can come up with that initial deposit.","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":34229,"title":"Calculation & Analysis","slug":"calculation-analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}}],"fromCategory":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282200,"slug":"finite-math-for-dummies","isbn":"9781119476368","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"amazon":{"default":"https://www.amazon.com/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119476364-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/finite-math-for-dummies-cover-9781119476368-203x255.jpg","width":203,"height":255},"title":"Finite Math For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <p><b data-author-id=\"8985\">Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-6217baa87271c\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-6217baa8730c2\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":254805}],"_links":{"self":{"self":"https://dummies-api.dummies.com/v2/categories/34229/categoryArticles?sortField=time&sortOrder=1&size=10&offset=0"},"next":{"self":"https://dummies-api.dummies.com/v2/categories/34229/categoryArticles?sortField=time&sortOrder=1&size=10&offset=10"},"last":{"self":"https://dummies-api.dummies.com/v2/categories/34229/categoryArticles?sortField=time&sortOrder=1&size=10&offset=206"}}},"objectTitle":"","status":"success","pageType":"article-category","objectId":"34229","page":1,"sortField":"time","sortOrder":1,"categoriesIds":[],"articleTypes":[],"filterData":{"categoriesFilter":[{"itemId":0,"itemName":"All Categories","count":216}],"articleTypeFilter":[{"articleType":"All Types","count":216},{"articleType":"Articles","count":213},{"articleType":"Cheat Sheet","count":3}]},"filterDataLoadedStatus":"success","pageSize":10},"adsState":{"pageScripts":{"headers":{"timestamp":"2022-05-13T12:59:08+00:00"},"adsId":0,"data":{"scripts":[{"pages":["all"],"location":"header","script":"<!--Optimizely Script-->\r\n<script src=\"https://cdn.optimizely.com/js/10563184655.js\"></script>","enabled":false},{"pages":["all"],"location":"header","script":"<!-- comScore Tag -->\r\n<script>var _comscore = _comscore || [];_comscore.push({ c1: \"2\", c2: \"15097263\" });(function() {var s = document.createElement(\"script\"), el = document.getElementsByTagName(\"script\")[0]; s.async = true;s.src = (document.location.protocol == \"https:\" ? 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Calculation & Analysis Articles

Whether you're trying to figure out the difference between covariance and correlation or breaking out a regression equation, our articles have the info you need.

Articles From Calculation & Analysis

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216 results
Calculation & Analysis Finite Math For Dummies Cheat Sheet

Cheat Sheet / Updated 03-08-2022

When performing the many types of computations found in Finite Math topics, it’s helpful to have some numbers, notations, distributions, and listings right at hand.

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Calculation & Analysis Reading Financial Reports For Dummies Cheat Sheet

Cheat Sheet / Updated 02-16-2022

If you're looking at a business with an interest in investing in it, you need to read its financial reports. Of course, when it comes to the annual report, you don't need to read everything, just the key parts. Combining the annual report with some of the financial reports a corporation files with the Securities and Exchange Commission (SEC) can help you figure profitability and liquidity ratios and get a better sense of cash flow. Keep this handy Cheat Sheet nearby for a quick reference to reading financial reports, including SEC reports, profitability ratios, liquidity ratios, and cash flow formulas.

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Calculation & Analysis Business Statistics For Dummies Cheat Sheet

Cheat Sheet / Updated 01-31-2022

Statistics make it possible to analyze real-world business problems with actual data so that you can determine if a marketing strategy is really working, how much a company should charge for its products, or any of a million other practical questions. The science of statistics uses regression analysis, hypothesis testing, sampling distributions, and more to ensure accurate data analysis.

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Calculation & Analysis Important Terms in Game Theory

Article / Updated 06-20-2019

As with many areas and topics in finite mathematics, there is a very special and specific vocabulary that goes along with game theory. Here are some important and useful terms that you should know. Payoff matrix: A matrix whose elements represent all the amounts won or lost by the row player. Payoff: An amount showing as an element in the payoff matrix, which indicates the amount gained or lost by the row player. Saddle point: The element in a payoff matrix that is the smallest in a particular row while, at the same time, the largest in its column. Not all matrices have saddle points. Strictly determined game: A game that has a saddle point. Strategy: A move or moves chosen by a player. Optimal strategy: The strategy that most benefits a player. Value (expected value) of game: The amount representing the result when the best possible strategy is played by each player. Zero-sum game: A game where what one player wins, the other loses; no money comes in from the outside or leaves. Fair game: A game with a value of 0. Pure strategy: A player always chooses the same row or column. Mixed strategy: A player changes the choice of row or column with different plays or turns. Dominated strategy: A strategy that is never considered because another play is always better. For the row player, a row is dominated by another row if all the corresponding elements are all larger. For the column player, a column is dominated by another column if all the corresponding elements are all smaller.

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Calculation & Analysis How to Create a Matrix from a Transition Diagram

Article / Updated 07-30-2018

When you encounter a matrix problem in finite math, a nice way to illustrate the transition from one state to another is to use a transition diagram. The different states are represented by circles, and the probability of going from one state to another is shown by using curves with arrows. The transition diagram in the following figure shows how an insurance company classifies its drivers: no accidents, one accident, or two or more accidents. This information could help the company determine the insurance premium rates. You see that 80% of the drivers who haven’t had an accident aren’t expected to have an accident the next year. Fifteen percent of those drivers have one accident, and 5% have two or more accidents. Seventy percent of those who have had one accident aren’t expected to have an accident the next year but have to stay in the one-accident classification. And those in the two-or-more accident class have to stay there. To create a transition matrix representing the drivers, use the percentages to show going from one state to another. What is the long-term expectation for these drivers? First, let the transition matrix be D. Then, some of the powers of D are At the end of ten years, using the drivers in the initial study, you have What this tells the insurance company is that, in ten years, about 11% of the original no-accident drivers will still not have had an accident. Only 3% of the one-accident drivers will still have had only that one accident. This situation doesn’t allow for the drivers to move back or earn forgiveness; a one-accident driver can’t be a no-accident driver using this model. Of course, different insurance agencies have different policies, putting drivers in better standing after a set number of accident-free years. And new policyholders are added to make this picture rosier. This just shows the pattern for a particular set of drivers after a certain number of years.

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Calculation & Analysis How to Use Transition Matrices

Article / Updated 07-30-2018

If your finite math instructor asks you to predict the likelihood of an action repeating over time, you may need to use a transition matrix to do this. A transition matrix consists of a square matrix that gives the probabilities of different states going from one to another. With a transition matrix, you can perform matrix multiplication and determine trends, if there are any, and make predications. Consider the table showing the purchasing patterns involving different cereals. You see all the percentages showing the probability of going from one state to another, but which of the cereals does the consumer actually end up buying most frequently in the long run? One way to look at continued purchasing is to create a tree diagram. In the following figure, you see two consecutive “rounds” of purchases. If you want the probability that the consumer purchases Kicks first, tries it again or something else, and then purchases Kicks the next time, add up the , , and branches: , or 38% of the time. If you want the probability that the consumer purchases Cheery A’s first, tries something else or repeats Cheery A’s, and then tries Corn Flecks, add up the , , and branches. This comes out to , or almost 26% of the time. The tree is helpful in that it shows you what the choices are and how the percentages work in determining patterns, but there’s a much easier and neater way to compute these values. To perform computations and study this further, create a transition matrix, referring back to the chart showing purchases and using the decimal values of the percentages. Name it matrix C. Next, use matrix multiplication to find C². As a quick hint, when multiplying matrices, you find the element in the first row, first column of the product, labeled c11, when you multiply the elements in the first row of the first matrix times the corresponding elements in the first column of the second matrix and then add up the products. In a matrix A, the element in the nth row, kth column is labeled ank. The element in the first row and second column of the product, c12, uses the elements in the first row of the first matrix and second column of the second matrix, and so on for the rest of the elements. So, you take the first row of the left matrix times the first column of the second matrix to get Yes. This is the same computation as was done using the tree to find the probability that a consumer starting with Kicks would return to it in two more purchases. Performing the matrix multiplication, you have Continuing this multiplication process, by the time C6 appears (the chances of buying a particular cereal at the fifth purchase time after the initial purchase), a pattern emerges. Notice that the numbers in each column round to the same three decimal places. This is going to become even clearer, using higher powers of C, until some nth matrix power becomes The matrix shows you the pattern or trend. No matter which cereal the consumer bought first, in the long run there’s a 35.3% chance that she’ll purchase Kicks, a 38.4% chance that she’ll purchase Cheery A’s, and a 26.3% chance that she’ll purchase Corn Flecks. This transition matrix has reached an equilibrium, where it won’t change with more repeated multiplication. You can write this situation with a single-line matrix:

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Calculation & Analysis How to Analyze Arguments with Euler Diagrams

Article / Updated 07-30-2018

On a finite math exam, you may be asked to analyze an argument with a visual approach using an Euler diagram. This pictorial technique is used to check to see whether an argument is valid. An argument can be classified as either valid or invalid. A valid argument occurs in situations where if the premises are true, then the conclusion must also be true. And an argument can be valid even if the conclusion is false. The following argument has two premises: (1) “All dogs have fleas.” (2) “Hank is a dog.” The conclusion is that, therefore, Hank has fleas. These arguments usually have the following format with the premises listed first and the conclusion under a horizontal line: Using an Euler diagram to analyze this argument, draw a circle to contain all objects that have fleas. Inside the circle, put another circle to contain all dogs. And inside the circle of dogs, put Hank. The figure illustrates this approach. The argument isn’t necessarily true, because you know that not all dogs have fleas. All this shows is that the argument is valid. If the two premises are true, then the conclusion must be true. Now consider an argument involving rectangles and triangles. A polygon is a figure made up of line segments connected at their endpoints. When analyzing the validity of this argument, the Euler diagram starts with a circle containing all polygons, as shown here. Two circles are drawn inside the larger circle—one containing rectangles and the other triangles. The two circles don’t overlap, because rectangles have four sides, and triangles have three sides. The argument is invalid. Rectangles are not triangles—not even sometimes. Arguments can have more than two premises. For example: One Euler diagram that can represent this situation has three intersecting circles, as shown here. As you can see from the diagram, there can be presidents born in Kentucky who were not lawyers in Illinois and there can be presidents who were lawyers in Illinois but not born in Kentucky. The argument is invalid. To be valid, it must always be true.

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Calculation & Analysis How to Analyze Compound Statements

Article / Updated 07-30-2018

If your finite math instructor asks you to analyze a compound statement, you can try using a truth table to do this. Not every topic in a discussion can be turned into a compound statement and analyzed for its truth that way, but using logic and truth values is a good technique to use when possible. Consider the compound statement When constructing a truth table, you start with the basic p and q columns. Then you add a ~ q column followed by a column Before you can perform the conjunction, ^, you need a ~ p column. Here’s a step-by-step procedure. Start with a basic p and q and then add ~ q. When adding the column, perform the disjunction on the first and third columns. Remember, with disjunctions, the statement is false only when both component statements are false. Add the ~ p column. Add the column, which shows the conjunction of the fourth and fifth columns. The conjunction is true only when the two component statements are true. This complex statement is only true when both original statements are false.

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Calculation & Analysis How to Calculate Monthly Payments for a Sinking Fund

Article / Updated 07-30-2018

A big part of finite math involves working through financial problems. Some of these problems may seem complex—like calculating the monthly deposits required to maintain a sinking fund. Fortunately, there’s a special formula you can use to find the answer. A sinking fund is usually used to accumulate money to fund a future expense or a way to retire a debt. You can use a sinking fund to pay off a loan in one lump sum at the end of a set amount of time while making just interest payments in the meantime. For example, a friend borrows $10,000 to purchase a boat and agrees to pay the full amount back in one payment, ten years from now. In the meantime, he agrees to pay interest monthly on the $10,000 at an annual rate of 12%. He also sets up a sinking fund to accumulate the lump-sum payment. The sinking fund earns 9% interest, compounded monthly. How much does he pay monthly? The monthly amount is both the interest to the lender and a deposit into the sinking fund. The interest to the lender is based on an annual rate of 12%. Using the simple interest formula, I = Prt, you have I = 10,000(0.12)(1) = 1,200 per year. Because he plans to make monthly payments, you divide by 12 so $100 per month goes for the interest payments. Next, you compute the amount to be deposited in the sinking fund each month. The formula for a sinking fund payment is where P is the amount of the payment, A is the amount to be accumulated, i is the interest rate per time period, and n is the number of time periods. Using the formula to determine the monthly payment into the sinking fund, the amount, A, is $10,000, and the interest per pay period is 9% divided by 12, because it’s compounded monthly. The number of time periods over the ten years is 120 So, the monthly payment into the sinking fund is about $51.68. Add that to the interest payments, and the monthly commitment is $151.68. In ten years, the monthly payments will end.

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Calculation & Analysis How to Calculate the Starting Value of an Annuity

Article / Updated 07-30-2018

In a finite math course, you’ll often be asked to use mathematical formulas to solve real-world problems. A good example of this is calculating the starting value of an annuity. Say that you’re planning an around-the-world-trip, and your big concern is how to fund this adventure. One way is to have a fund from which you can withdraw a certain amount of money at regular intervals and have just enough money when you’re finished, where the fund goes down to zero. You’re going to sail around the world on your 40-foot sloop. You’re estimating it will take seven years, with all the visiting, sightseeing, and other activities. You need to set up an annuity from which you can withdraw monthly amounts to help with the expenses. You want to have $2,000 available each month and have the balance be zero at the end of the seven years. How much should you put in your annuity account? The present value of an annuity is determined with where V is the value or amount needed to be deposited into the account, P is the payment or amount withdrawn periodically, i is the interest each time period, and m is the number of time periods. You find a broker who can get you 9% interest on your deposit. You want to withdraw monthly, so that will be 7 times 12, or 84 payments or time periods. The interest rate each time period is found with r/n, which is 9% divided by 12, or 0.0075%. Using the formula for the present value, that comes to You need to deposit more than $124,000 to be able to make your regular withdrawals and have a balance of zero at the end of seven years. What is $2,000 per month for seven years, if you aren’t withdrawing from an annuity? Multiply Sounds like a good deal—if you can come up with that initial deposit.

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