EXAMPLE 3 Pair-a-Dice: Outcomes for Rolling Two Dice

Rolling one six-sided die has an obvious sample space of six equally likely outcomes: . But many board games (and casino games) involve rolling two dice and noting the sum of the spots on the two sides that are facing up. We know from our experience playing games like Monopoly that the 11 possible sums are not equally likely because, for example, there are many ways to get a sum of 7 but only one way to get a sum of 12.

We start by identifying the sample space in order to find the exact probabilities and patterns of the various dice sums. Because of the large number of possible outcomes, the table in Figure 8.7 is a more straightforward representation of the sample space 5 than a tree diagram would be. Figure 8.7 shows possible (and equally likely) ways to roll two dice.

The longest rising diagonal of the table shows the six ways that the sum can equal 7. Therefore, it makes sense that the probability of the sum being 7 is .