[9 marks]. Arrivals of the Number 1 bus at a particular bus stop form a Poission process with rate parameter 1 and arrivals of the Number 2 bus at the same bus stop form a Poisson process with rate parameter 2. The two Poisson processes are independent.

(a) What is the probability that precisely 4 buses (of type Number 1 and/or Number 2) arrive in the time interval [0, t]?

(b) What is the probability that exactly 3 Number 2 buses pass while I am waiting for a Number 1 bus?

(c) Whenthemaintenancedepotgoesonstrikehalfthebusesbreakdownbeforetheyreach my stop. What then is the probability that not a single bus passes in the time interval [0, t].

When answering each of parts (a), (b) and (c) you should briefly explain your reasoning.

(ii) In part (c) you may assume that thinning has taken place, as defined in the lecture notes.