Difficult riddle about impatient people

[americanpsy]

Two impatient people agree to meet up sometime between 3pm and 4 pm, in the era before mobile phones, just near Nelson's Column in London. They are so impatient they will only wait 15 minutes before leaving. What is the probability they actually meet up?

 

[Rogerio]

Supposing nobody will arrive before 3pm or after 4pm...

Spoiler

7/16

 

[americanpsy]

Yes that is the assumption. No arrivals before 3pm or after 4pm.

 

[Rogerio]

An easy way to understand the answer...

 

[mfb]

If it is common knowledge that they will not arrive before 3 pm or after 4 pm, and wait for 15 minutes, they can do better: It is not necessary to show up before 3:15 - the other one will not have left before that. In the same way, it is not necessary to be there after 4 pm. Assuming that just one of them comes to that conclusion and the other one arrives at a random time, you can get a meeting probability of 1/2.
If both come to that conclusion, but choose a random arrival time between 3:15 and 3:45, their meeting probability is 3/4.
If they both know that the other one will do this analysis, they are both there around 3:30 and the meeting probability is 1.