Probability

1 - (1-1/4)^2 = 7/16

For any given time that Joan shows up, there is a thirty minute window for Jim's time where he would meet her. He can be up to 15 minutes earlier and be waiting or he can be up to 15 minutes later and she will wait for him. This window is shorter if close to the bounds. If Joan shows up exactly at 3:00, there is only a 15 minute window. So from 3:00 to 3:15, the window goes linearly from 15 minutes to 30 minutes, from 3:15 to 3:45 it is 30 minutes and from 3:45 to 4:00 is drops linearly from 30 to 15 minutes. The average window time is 26.25 minutes divided by 60 minutes total is 43.75%

Mark, we clearly got the same answer, but yours looks so elegant and I don't see how you got there. Can you walk through your thinking?

Graph the overlap with time for one person on the x axis and time for the other person on the y axis. You end up with a 1x1 square with a "thick" diagonal from (0,0) to (1,1). The diagonal is bounded by two line segments: (0,0.25)-(0.75,1) and (0.25,0)-(1,0.75). That leaves two triangular sections that can be combined to create a 0.75x0.75 square. Subtract the small square from the 1x1 square to get the area of the overlap.

Nice. I like the graphical approach.

We assume that each indeed arrives at some time within the hour. Each has a 15 minute waiting period.

1. If the first person to arrive comes at 3:45 or later they will surely meet and the probability of that is 1/4.

2. If the first person to arrive comes before 3:45 (p = 3/4) they will meet only if the next person comes within the next 15 minutes (p = 1/4) for a total probability of 3/16.

Either 1 or 2 happens but not both, so the total p is 7/16. Smart guys!