3. [Binomial Passenger Trains] Suppose that Amtrak constructs northbound trains using the following algorithm. The train-yard in Memphis has six train cars. If a car works, it is added to the northbound train; if not, it is left in Memphis. Each car works with a probability of p = 2 3 . (a) What is the expected length, in cars, of a northbound train? (b) What is the probability of a length-zero train? (c) The number of passengers who can ride on a northbound train is 50X, where X is the number of train cars. The number of passengers who want to ride the train, Y , is a uniformly distributed random variable, pY (v) = 1 100 for 101 v 200. What is the probability that Amtrak will be able to carry all of the passengers who wish to ride