Poisson processes can effectively model the arrival of shocks to a system (e.g., disruptions in a financial system, physical phenomena, surges in demand, etc.). Suppose that we model the arrival of shocks to a system as a Poisson process with a rate of = 2 shocks per hour.

(1) The system starts at time t = 0. Calculate the probability that exactly three shocks occur by time t = 1.

(2) The system experiences 7 shocks over the course of five hours. Given this, calculate the probability that exactly three of the shocks occurred in the first four hours. (

3) Calculate the probability that the system experiences exactly one shock between time t = 0 and time t = 1 and three shocks between time t = 3 and time t = 6.