Random processes include rolling a die and flipping a coin. (a) Think of another random process. (b) Describe all the possible outcomes of that process. For instance, rolling a die is a random process with possible outcomes 1, 2, …, 6.1
What we think of as random processes are not necessarily random, but they may just be too difficult to understand exactly. The fourth example in the footnote solution to Guided Practice 3.6 suggests a roommate’s behavior is a random process. However, even if a roommate’s behavior is not truly random, modeling her behavior as a random process can still be useful.
Disjoint or mutually exclusive outcomes
Two outcomes are called disjoint or mutually exclusive if they cannot both happen. For instance, if we roll a die, the outcomes 1 and 2 are disjoint since they cannot both occur. On the other hand, the outcomes 1 and “rolling an odd number” are not disjoint since both occur if the outcome of the roll is a 1. The terms disjoint and mutually exclusive are equivalent and interchangeable.
Calculating the probability of disjoint outcomes is easy. When rolling a die, the outcomes 1
and 2 are disjoint, and we compute the probability that one of these outcomes will occur by adding their separate probabilities:
P (1 or 2) = P (1) + P (2) = 1/6 + 1/6 = 1/3
What about the probability of rolling a 1, 2, 3, 4, 5, or 6? Here again, all of the outcomes are disjoint so we add the probabilities:
P (1 or 2 or 3 or 4 or 5 or 6)
= P (1) + P (2) + P (3) + P (4) + P (5) + P (6)
= 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1
The Addition Rule guarantees the accuracy of this approach when the outcomes are disjoint.