Cars pass an intersection according to a Poisson process with rate . There are 4 types of

Cars pass an intersection according to a Poisson process with rate λ. There are 4 types of cars, and each passing car is, independently, type i with probability pi , 4 i=1 pi = 1.

(a) Find the probability that at least one of each of car types 1, 2, 3 but none of type 4 have passed by time t .

(b) Given that exactly 6 cars of type 1 or 2 passed by time t , find the probability that 4 of them were type 1.