Chapter 12.5, Problem 8E

Number of Cars at a Tollgate During a certain part of the day, an average of five automobiles arrives every minute at the tollgate on a turnpike. Let $X$ be the number of automobiles that arrive in any $1$-minute interval selected at random. Let $Y$ be the interarrival time between any two successive arrivals. (The average interarrival time is $15$ minute.) Assume that $X$ is a Poisson random variable and that $Y$ is an exponential random variable

a. Find the probability that at least five cars arrive during a given $1$-minute interval.

b. Find the probability that the time between any two successive cars is less than $\frac{1}{5}$ minute.