A man and a woman agree to meet at a cafe about noon If the

A man and a woman agree to meet at a cafe about noon. If the man arrives at a time uniformly distributed between 15 minutes? 12:25 , what is the probability that the first to arrive waits no longer than 11:50 and 12:10 and if the woman independently arrives at a time uniformly distributed between 11:35 and

Solution

Let X be the number of minutes past noon that the man arrives, and let Y represent the number of minutes past noon that the woman arrives. From the problem, we know that 0<X<45and 0<Y<35. First we find the probability that the first to arrive waits no longer than 5 minutes for the other person. There are two ways to do this: either the man can arrive first and waits for the woman, or vice-verse.

First consider the situation that the man arrives first. The man arrives sometime between 35 minutespast 11 and 10 minutes past noon. The acceptable time for the woman to arrive is no later than 15 minutes after the man. Given that the man arrives between 35 minute after 11 and 10 minutes after noon, the woman independently arrives some time within the hour. This area is shaded in light grey and has area 1575.So that the man does not get impatient, the woman must arrive before 5 minutes have passed. The minutes for the man and woman to arrive such that the man does not become impatient is shaded in black and has area 525 . The probability that the woman arrives no more than 5 minutes after the man is 525/1575= 1/3