2. 7 Bayes\" Theorem EX.1 Suppose that Bob can decide to go to work by one of three modes of transportation: ear, bus, and commuter train. Because of high trafc, if he decides to go by car, there is a 50% chance that he will be late. If he goes by bus, which has reserved lanes, but is sometimes overcrowded, the probability of being late is only 20%. The commuter train is almost never late, with a probability of only.r 1%, but is more expensive than the bus. [a] Suppose that Bob is late one day, and his boss wishes to estimate the probability that he drove to work that day in a car. Sinec he does not know which mode of transportation Bob usually uses, he gives a prior probability of % to each of the three possibilities. What is the boss' estimate of the probability that Bob drove to work? (b) Suppose that a coworker of Bob's knows that he almost always takes the commuter train to work, never takes the bus, but sometimes, 10% of the times, takes the car to work. What is the coworkers probability that Bob drove to work that day, given that he was late