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!