8.5 8.4 Discrete Probability Models

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.

Table 8.6: Table 8.5 Probability Model for Rolling a Pair of Dice Until You Get Doubles
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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Self Check 7

We continue with the game of Monopoly and of finding probabilities associated with getting out of jail.

  1. The official rules allow a jailed player to try for doubles on three consecutive turns. If after three tries the player does not roll a double, then the player must pay $50 to get out of jail. What is the probability that the player will be able to get out of jail without paying a fine?

  2. Return to Example 11. After a Donovan family member was stuck in jail for 10 turns, the Donovans changed their rules so that jailed players get out of jail only after rolling a sum less than 7. Write a probability model for getting out of jail under the Donovans’ new rule.

    • Number of
      rolls until
      sum 7
      1 2 3 4 5
      Probability

  3. Does the Donovans’ new rule make it less likely that a player will spend a long time in jail? As part of your answer, calculate the probability that a jailed player gets out of jail within his or her first three rolls.

    • This rule makes it less likely that players will spend a long time in jail. , which is nearly twice the probability under the Donovans’ old rules.

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).

Table 8.8: Table 8.6 Probability Model Known as Benford’s Law
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.

Self Check 8

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).

  1. Determine the proportion of your fake invoices that have 1, 2, or 3 as the first digit in their invoice numbers.

    • 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.

  2. Compare your answer in part (a) to the probability of observing an invoice number with a first digit of 1, 2, or 3 based on Benford’s law (Table 8.6). Do you think your fraud will be detected?

    • The probability of observing a 1, 2, or 3, according to Benford’s law is 0.602. That is quite a bit higher than the answer to part (a); hence our fraud is likely to be detected.