1. Traffic at a certain intersection follows a Poisson process with rate 2/3 vehicles per minute. Ten percent of the vehicles are trucks; the other ninety percent are cars.

a) What is the probability at least one truck passes in an hour?

b) Given that 10 trucks have passed by in an hour, what is the expected number of vehicles that have passed by?

c) Given that 50 vehicles have passed by in an hour, what is the probability there were exactly 5 trucks and 45 cars?

2.Hockey teams 1 and 2 score goals at times of Poisson process with rates 1 and 2. Suppose that at the beginning the score is 3 to 1 (team 1 has 3 goals, and team 2 has 1 goal).

a) What is the probability that ????₁(????) will reach 5 before ????₂(????) does?

b) Now, repeat a) if the Poisson processes have rates ????₁ and ????₂.