Suppose you are taking the bus home, and can take either the 51 or the 82. you know that the 82 will arrive in exactly 10 minutes, but because the 51 is unreliable, you suspect that the amount of time until it arrives is uniformly distributed between 0 and 30 minutes. you take the first bus that arrives. let t be the number of minutes you wait until you board a bus. find e[t].

We are dealing here with a uniform distribution ranging from 0 to 30 minutes.  We need to calculate the probability that the unreliable bus will arrive before the reliable one.  This probability is the area under the uniform distribution "curve" from 0 to 10 minutes.  This constitutes 1/3 of the entire unform distr. curve.  So the probability that the unreliable bus will arrive before the reliable one is 1/3, or 0.33.  The probability that it will arrive AFTER the reliable bus is 2/3, or 0.67.