Calculation & Analysis Articles

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.

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.

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.

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.

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.

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:

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.

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.

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.

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.