In this chapter, we will work with two kinds of probability models: discrete probability models and continuous probability models. The probability models in Tables 8.3 (page 353) and 8.4 (page 358) are examples of the first kind. In both cases, the number of possible outcomes—the sums from two dice (Table 8.3) or the test results (Table 8.4)—is finite, hence the outcomes can be listed. If all the outcomes in a sample space can be put into a list, the number of outcomes is said to be countable. A probability model for which the sample space is countable is called a discrete probability model.
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Discrete Probability Model DEFINITION
A discrete probability model is a probability model with a countable number of outcomes in its sample space.
To assign probabilities in a discrete model, list the probability of all the individual outcomes. By Rules 1 and 2, these probabilities must be numbers between 0 and 1 inclusive and must sum to 1. The probability of any event is the sum of the probabilities of the outcomes making up the event.
Up to this point, all the probability models discussed have finite sample spaces. But, as you will see in Example 11, that is not always the case.
EXAMPLE 11 Probability Model: Rolling a Pair of Dice Until You Get Doubles
According to the Donovan family’s custom rules for Monopoly, if you land on the “Go to jail” square, the only way to get out of jail is to roll doubles. Let represent rolling doubles and represent any roll that does not result in doubles. The sample space for this situation is . In this case, the sample space contains an infinite number of outcomes, which can be put into a list. To form a discrete probability model, we need to assign probabilities to each outcome in the list. Here are calculations for some of the probabilities:
Algebra Review Appendix
Powers and Roots Operations with Rational Numbers
Continuing this pattern, we can form a probability model in which we list the possible outcomes (number of rolls needed to get doubles) and their corresponding probabilities. Table 8.5 shows this probability model.
| Number of rolls until doubles | 1 | 2 | 3 | 4 | 5 | … | … | |
| Probability | … | … |
It takes a bit of work to show that the probabilities sum to 1, but they do! So, this is a valid probability model.
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We continue with the game of Monopoly and of finding probabilities associated with getting out of jail.
| Number of rolls until sum 7 |
1 | 2 | 3 | 4 | 5 | … | … | |
|---|---|---|---|---|---|---|---|---|
| Probability | … | … |
Example 12 gives another example of a discrete probability model, but this time the sample space is finite.
EXAMPLE 12 Benford’s Law: One Is the Likeliest Number You’ll Ever Know
Faked numbers in tax returns, invoices, or expense account claims often display patterns that aren’t present in legitimate records. Some patterns, like too many round numbers, are obvious and easily avoided by a clever crook. Others are more subtle. It is a striking fact that the first (leftmost) digits of numbers in legitimate records often follow a model known as Benford’s law, which is shown in Table 8.6. (Note that a first digit can’t be 0).
| First digit | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Probability | 0.301 | 0.176 | 0.125 | 0.097 | 0.079 | 0.067 | 0.058 | 0.051 | 0.046 |
You should check that the probabilities of the outcomes sum exactly to 1 to verify that this is a legitimate discrete probability model. Using this model, investigators can detect fraud by comparing the first digits in records such as invoices paid by a business with these probabilities. For example, consider the events = “first digit is 1” and = “first digit is 2.” Applying Rule 4, the addition rule for disjoint events, to the table of probabilities yields , which is 0.477 (almost 50%). Crooks trying to “make up” the numbers probably would not make up numbers starting with 1 or 2 this often.
You decide to fake 20 invoices. To make sure that you don’t introduce any pattern into your invoice numbers, you randomly assign numbers. Use Table 7.1, the random digits table (page 298), to assign the first digits to 20 fake invoices. Enter the table on line 106 (skip any 0s).
Using the table, the first digits on the invoice numbers are:
The proportion of numbers assigned a first digit of 1, 2, or 3 is 0.35.