A local law enforcement agency claims that the number of times that a patrol car passes through a particular neighborhood follows a Poisson process with a mean of three times per nightly shift. Let x denote the number of times that a patrol car passes through the neighborhood during a nightly shift.
a. Calculate the probability that no patrol cars pass through the neighborhood during a nightly shift.
b. Suppose that during a randomly selected night shift no patrol cars pass through the neighborhood. Based on your answer in part a, do you believe the agency's claim? Explain.
c. Assuming that nightly shifts are independent and assuming that the agency's claim is correct, find the probability that exactly one patrol car will pass through the neighborhood on each of four consecutive nights.