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.