What is the probability of two people meeting in a bar for exactly 15 minutes?

[Ganesh Ujwal]

Two people have to spend exactly 15 consecutive minutes in a bar on a given day, between 12:00 and 13:00. Assuming uniform arrival times, what is the probability they will meet?I am mainly interested to see how people would model this formally. I came up with the answer 50% (wrong!) based on the assumptions that:- independent uniform arrival
- they will meet iff they actually overlap by some $$ ϵ>0$$
- we can measure time continuouslybut my methods felt a little ad hoc to me, and I would like to learn to make it more formal.Also I'm curious whether people think the problem is formulated unambiguously. I added the assumption of independent arrival myself for instance, because I think without such an assumption the problem is not well defined.

 

[jedishrfu]

MAth Stack Exchange has a great discussion on this very problem:

https://math.stackexchange.com/questions/103015/chance-of-meeting-in-a-bar#103035

They solve the problem graphically and note a few restrictions such as the arrival times for the two people are between 12:00 and 12:45 since both are required to be at the bar for 15 minutes.