Probability For Dummies book cover

Probability For Dummies


Packed with practical tips and techniques for solving probability problems

Increase your chances of acing that probability exam -- or winning at the casino!

Whether you're hitting the books for a probability or statistics course or hitting the tables at a casino, working out probabilities can be problematic. This book helps you even the odds. Using easy-to-understand explanations and examples, it demystifies probability -- and even offers savvy tips to boost your chances of gambling success!

Discover how to
* Conquer combinations and permutations
* Understand probability models from binomial to exponential
* Make good decisions using probability
* Play the odds in poker, roulette, and other games
Packed with practical tips and techniques for solving probability problems

Increase your chances of acing that probability exam -- or winning at the casino!

Whether you're hitting the books for a probability or statistics course or hitting the tables at a casino, working out probabilities can be problematic. This book helps you even the odds. Using easy-to-understand
explanations and examples, it demystifies probability -- and even offers savvy tips to boost your chances of gambling success!

Discover how to
* Conquer combinations and permutations
* Understand probability models from binomial to exponential
* Make good decisions using probability
* Play the odds in poker, roulette, and other games
Probability For Dummies Cheat Sheet

Successfully working your way through probability problems means understanding some basic rules of probability along with discrete and continuous probability distributions. Here are some helpful study tips to help you get well-prepared for a probability exam.

Articles From The Book

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Statistics Articles

Using Probability When Hitting the Slot Machines

Remember the movie National Lampoon's Vegas Vacation, when gambling fever consumes Chevy Chase's character, Clark W. Griswold? He goes on a losing streak to beat all losing streaks while his son, Rusty, wins four cars by playing the slot machines. Maybe Clark would have done better if he had read Probability For Dummies. In this article, you'll discover the basics of slot machines and how they work, so that you can get past the myths and develop a sound strategy based on probability.

Understanding average payout

When casinos advertise that their slot machines pay out an average of 90 percent, the fine print they don't want you to read says that you lose 10 cents from each dollar you put into the machines in the long term. (In probability terms, this means that your expected winnings are minus 10 cents on every dollar you spend every time the money goes through the machines.)

Suppose you start with $100 and bet a dollar at a time, for example. After inserting all $100 into the slot, 100 pulls later, you'll end up on average with $90, because you lose 10 percent of your money. If you run the $90 back through the machine, you'll end up with 90 percent of it back, which is 0.90 x 90 = $81. If you run that amount through in 81 pulls, you'll have $72.90 afterward (0.90 x 81 = 72.90). If you keep going for 44 rounds, on average, the money will be gone (unless you have the luck of Rusty Griswold). How many pulls on the machine does your $100 give you at this rate? Each time you have less money to run through the machine, so you have fewer pulls left. If you insert $1 at a time, you can expect 972 total pulls in the long term with these average payouts (that's the total pulls in 44 rounds). But keep in mind that casinos are designing slot machines to go faster and faster between spins. Some are even doing away with the handles and tokens by using digital readouts on gaming cards that you put into the machines. The faster machines can play up to 25 spins per hour, and 972 spins divided by 25 spins per minute is 38.88 minutes. You don't have a very long time to enjoy your $100 before it's gone!

The worst part? Casinos often advertise that their "average payouts" are as high as 95 percent. But beware: That number applies only to certain machines, and the casinos don't rush to tell you which ones. You really need to read or ask about the fine print before playing. You can also check the information on the machine to see if it lists its payouts. (Don't expect this information to be front and center.)

It's best to have a strategy

Advice varies regarding whether you should play nickel, quarter, or dollar slot machines and whether you should max out the number of coins you bet or not. (You usually get to choose between one and five coins to bet on a standard slot machine.) Here are a few tips for getting the most bang for your buck (or nickel) when playing slot machines.

Basically, when it comes to slot machines, strategy boils down to this: Know the rules, your probability of winning, and the expected payouts; dispel any myths; and quit while you're ahead. If you win $100, cash out $50 and play with the rest, for example. After you lose a certain amount (determined by you in advance), don't hesitate to quit. Go to the all-you-can-eat buffet and try your luck with the casino food; odds are it's pretty good.

Choosing among nickel, quarter, and dollar machines

The machines that have the higher denominations usually give the best payouts. So, between the nickel and quarter slots, for example, the quarter slots generally give better payouts. However, you run the risk of getting in way over your head in a hurry, so don't bet more than you can afford to lose. The bottom line: Always choose a level that you have fun playing at and that allows you to play for the whole time you've allotted.

Deciding how many coins to play at a time

When deciding on the number of coins you should play per spin, keep in mind that more is sometimes better. If the slot machine gives you more than twice the payout when you put in twice the number of coins, for example, you should max it out instead of playing single coins, because you increase your chances of winning a bigger pot, and the expected value is higher. If the machine just gives you k times the payout for k coins, it doesn't matter if you use the maximum number of coins. You may as well play one at a time until you can make some money and leave so your money lasts a little longer. For example, say a quarter machine pays 10 credits for the outcome 777 when you play only a single quarter, but if you play two quarters, it gives you 25 credits for the same outcome. And if you play the maximum number of quarters (say, four), a 777 results in 1,000 credits. You can see that playing four quarters at a time gives you a better chance of winning a bigger pot in the long run (if you win, that is) compared to playing a single quarter at a time for four consecutive tries.

What about penny slot machines? Although these profess to require only a penny for a spin, you get this rate only if you want to bet one penny at a time. The machines entice you to bet way more than one penny at a time; in fact, on some machines, you can bet more than 1,000 coins (called lines) on each spin — $10 a shot here, folks. Because these machines take any denomination of paper bill, as well as credit cards, your money can go faster on penny machines than on dollar machines because you can quickly lose track of what you spend. Pinching pennies may not be worth it after all.

Statistics Articles

Figuring Out What Probability Means

Probabilities come in many different disguises. Some of the terms people use for probability are chance, likelihood, odds, percentage, and proportion. But the basic definition of probability is the long-term chance that a certain outcome will occur from some random process. A probability is a number between zero and one — a proportion, in other words. You can write it as a percentage, because people like to talk about probability as a percentage chance, or you can put it in the form of odds. The term "odds," however, isn't exactly the same as probability. Odds refers to the ratio of the denominator of a probability to the numerator of a probability. For example, if the probability of a horse winning a race is 50 percent (1/2), the odds of this horse winning are 2 to 1.

Understanding the concept of chance

The term chance can take on many meanings. It can apply to an individual ("What are my chances of winning the lottery?"), or it can apply to a group ("The overall percentage of adults who get cancer is . . ."). You can signify a chance with a percent (80 percent), a proportion (0.80), or a word (such as "likely"). The bottom line of all probability terms is that they revolve around the idea of a long-term chance. When you're looking at a random process (and most occurrences in the world are the results of random processes for which the outcomes are never certain), you know that certain outcomes can happen, and you often weigh those outcomes in your mind. It all comes down to long-term chance; what's the chance that this or that outcome is going to occur in the long term (or over many individuals)?

If the chance of rain tomorrow is 30 percent, does that mean it won't rain because the chance is less than 50 percent? No. If the chance of rain is 30 percent, a meteorologist has looked at many days with similar conditions as tomorrow, and it rained on 30 percent of those days (and didn't rain the other 70 percent). So, a 30 percent chance for rain means only that it's unlikely to rain.

Interpreting probabilities: Thinking large and long-term

You can interpret a probability as it applies to an individual or as it applies to a group. Because probabilities stand for long-term percentages, it may be easier to see how they apply to a group rather than to an individual. But sometimes one way makes more sense than the other, depending on the situation you face. The following sections outline ways to interpret probabilities as they apply to groups or individuals so you don't run into misinterpretation problems.

Playing the instant lottery

Probabilities are based on long-term percentages (over thousands of trials), so when you apply them to a group, the group has to be large enough (the larger the better, but at least 1,500 or so items or individuals) for the probabilities to really apply. Here's an example where long-term interpretation makes sense in place of short-term interpretation. Suppose the chance of winning a prize in an instant lottery game is 1/10, or 10 percent. This probability means that in the long term (over thousands of tickets), 10 percent of all instant lottery tickets purchased for this game will win a prize, and 90 percent won't. It doesn't mean that if you buy 10 tickets, one of them will automatically win.

If you buy many sets of 10 tickets, on average, 10 percent of your tickets will win, but sometimes a group of 10 has multiple winners, and sometimes it has no winners. The winners are mixed up amongst the total population of tickets. If you buy exactly 10 tickets, each with a 10 percent chance of winning, you might expect a high chance of winning at least one prize. But the chance of you winning at least one prize with those 10 tickets is actually only 65 percent, and the chance of winning nothing is 35 percent.

Pondering political affiliation

You can use the following example as an illustration of the limitation of probability — namely that actual probability often applies to the percentage of a large group. Suppose you know that 60 percent of the people in your community are Democrats, 30 percent are Republicans, and the remaining 10 percent are Independents or have another political affiliation. If you randomly select one person from your community, what's the chance the person is a Democrat? The chance is 60 percent. You can't say that the person is surely a Democrat because the chance is over 50 percent; the percentages just tell you that the person is more likely to be a Democrat. Of course, after you ask the person, he or she is either a Democrat or not; you can't be 60-percent Democrat.

Seeing probability in everyday life

Probabilities affect the biggest and smallest decisions of people's lives. Pregnant women look at the probabilities of their babies having certain genetic disorders. Before you sign the papers to have surgery, doctors and nurses tell you about the chances that you'll have complications. And before you buy a vehicle, you can find out probabilities for almost every topic regarding that vehicle, including the chance of repairs becoming necessary, of the vehicle lasting a certain number of miles, or of you surviving a front-end crash or rollover (the latter depends on whether you wear a seatbelt — another fact based on probability).

Here are a couple of examples of probabilities that affect people's everyday lives:

  • Distributing prescription medications in specially designed blister packages rather than in bottles may increase the likelihood that consumers will take the medication properly, a new study suggests. (Source: Ohio State University Research News, June 20, 2005)
    In other words, the probability of consumers taking their medications properly is higher if companies put the medications in the new packaging than it is when the companies put the medicines in bottles. You don't know what the probability of taking those medications correctly was originally or how much the probability increases with this new packaging, but you do know that according to this study, the packaging is having some effect.
  • According to State Farm Insurance, the top three cities for auto theft in Ohio are Toledo (580.23 thefts per 100,000 vehicles), Columbus (558.19 per 100,000), and Dayton-Springfield (525.06 per 100,000).
    The information in this example is given in terms of rate; the study recorded the number of cars stolen each year in various metropolitan areas of Ohio. Note that the study reports the information as the number of thefts per 100,000 vehicles. The researchers needed a fixed number of vehicles in order to be fair about the comparison. If the study used only the number of thefts, cities with more cars would always rank higher than cities with fewer cars.
    How did the researchers get the specific numbers for this study? They took the actual number of thefts and divided it by the total number of vehicles to get a very small decimal value. They multiplied that value by 100,000 to get a number that's fair for comparison. To write the rates as probabilities, they simply divided them by 100,000 to put them back in decimal form. For Toledo, the probability of car theft is 580.23/100,000 = 0.0058023, or 0.58 percent; for Columbus, the probability of car theft is 0.0055819, or 0.56 percent; and for Dayton-Springfield, the probability is 0.0052506, or 0.53 percent.

Be sure to understand exactly what format people use to discuss or report a probability, and be sure that the format allows for a fair and equitable comparison.

Statistics Articles

Discrete Probability Distributions

In probability, a discrete distribution has either a finite or a countably infinite number of possible values. That means you can enumerate or make a listing of all possible values, such as 1, 2, 3, 4, 5, 6 or 1, 2, 3, . . .

There are several kinds of discrete probability distributions, including discrete uniform, binomial, Poisson, geometric, negative binomial, and hypergeometric.