Problem 5: A traffic control engineer believes that the cars passing through a particular intersection arrive at a mean rate equal to either 3 or 5 for a given time interval.

Prior to collecting any data, the engineer believes that it is much more likely that the rate λ = 3 than λ = 5. In fact, the engineer believes that the prior probabilities are: P(λ = 3) = 0.7 and P(λ = 5) = 0.3 One day, during a randomly selected time interval, the engineer observes x = 7 cars pass through the intersection. In light of the engineer's observation, what is the probability that λ = 3? And what is the probability that λ = 5?