EXAMPLE 2 Rolling two dice

Rolling two dice is a common way to lose money in casinos. There are 36 possible outcomes when we roll two dice and record the up-faces in order (first die, second die). Figure 18.1 displays these outcomes. What probabilities should we assign?

Casino dice are carefully made. Their spots are not hollowed out, which would give the faces different weights, but are filled with white plastic of the same density as the red plastic of the body. For casino dice, it is reasonable to assign the same probability to each of the 36 outcomes in Figure 18.1. Because these 36 probabilities must have sum 1 (Rule B), each outcome must have probability 1/36, or 1-in-36.

We are interested in the sum of the spots on the up-faces of the dice. What is the probability that this sum is 5? The event “roll a 5” contains four outcomes, and its probability is the sum of the probabilities of these outcomes: