Consider a collection A1, . . . , Ak of mutually exclusive and exhaustive events, and a random variable X whose distribution depends on which of the Ai

’s occurs (e.g., a commuter might select one of three possible routes from home to work, with X representing the commute time). Let denote the expected value of X given that the event Ai occurs. Then it can be shown that

, the weighted average of the individual “conditional expectations” where the weights are the probabilities of the partitioning events.

a. The expected duration of a voice call to a particular telephone number is 3 minutes, whereas the expected duration of a data call to that same number is 1 minute.

If 75% of all calls are voice calls, what is the expected duration of the next call?

b. A deli sells three different types of chocolate chip cookies. The number of chocolate chips in a type i cookie has a Poisson distribution with parameter

. If 20% of all customers purchasing a chocolate chip cookie select the first type, 50% choose the second type, and the remaining 30%

opt for the third type, what is the expected number of chips in a cookie purchased by the next customer?