6. (15 points) Cars pass an intersection according to a Poisson process with rate A = 10 cars per minute. There are 3 types of cars, and each passing car is, independently, type i with probability pg, 73 E {1, 2, 3}, where 101 = 0-3, p2 = 0.5, p3 = 0.2. Let time zero denote the instant at which we sit down and begin observing cars passing by at the intersection. (a) Find the probability that at least one car of Type 1, and at least one car of Type 2 has passed by time t (measured in minutes) but no cars of Type 3 have passed by time t. (b) Suppose you know that exactly 5 cars of Type 1 or 2 have passed by time t. What is the probability exactly 3 were of Type 1? (c) Suppose we leave as soon as we see a Type 3 car pass the intersection. Calculate the expected number of cars of Type 1 or 2 we observe passing the intersection before we leave