You are walking towards a shuttle stop when you see the shuttle pull away. you run towards the shuttle waving your arms, hoping that the driver sees you. let’s say there is a fifty percent chance that the driver sees you. if the driver does not see you, then you walk home, which takes you exactly 20 minutes. if the driver sees you, you get on the shuttle, which takes a random amount of time (depending on traffic) to take you home. let x be the amount of time it takes you to get home.

(a) suppose that the amount of the time it takes the shuttle to get you home is a uniform random variable between 5 and 15 minutes. find e[x] using the total expectation theorem.

(b) suppose that the amount of the time it takes the shuttle to get you home is distributed as exp(1/8) (this is probably not a good model for traffic). find e[x] using the total expectation theorem.