Tom enters the post office where 5 people are being served, each by a different sales clerk. He will be called up as soon as any one of the 5 people currently being attended to are finished. The service time for each individual by each cleark has an exponential distribution with an average service time of 5 minutes, and is independent of all other service times. Find the probability Tom has to wait for more than 2 minutes before he is called up.
I'm struggling with determining how to set this up, mainly with the fact that there are 5 people being served.
For Tom to wait more than 2 minutes, each of the 5 clerks will have to take more than 2 minutes on their respective customers. So if x is the probability that a single clerk will take longer than 2 minutes (I'll let you compute x), then the final answer would just be x to the power 5. This is a joint probability distribution. P(tom waits longer than 2 minutes) = P(clerk 1 takes longer than 2 minutes,clerk 2 takes longer than 2 minutes,etc.etc.) = P(a single clerk takes longer than 2 minutes)^5.