Suppose that you can go to work with any of three different transportation's; car, bus or train. However, your decision is greatly affected by the traffic capacity. In case of high traffic, if you decide to go by car, there is a 50% chance to be late but if you choose to go by bus, the probability of being late is only 20%. The train is almost never late, with a probability of only 1%, but its more expensive than the bus.

a. Suppose that you are late one day, and your boss wishes to estimate the probability that you drove to the work on that day by car. Since he does not know which mode of transportation you usually use. The boss gives a prior probability of 1/3 to each of the three possibilities. What is the boss`s estimate of the probability that you drove to work?

b. Suppose that this boss has previous knowledge that you almost always take the train to the work, never takes the bus, but sometimes, 10% of the time, you take the car. What is the boss`s probability that you drove to work that day, given that you were late?