# Probability of two people meeting during a certain time.

I recently read a math problem and, having not yet taken anything beyond calculus 1, was curious about how to solve it correctly.

Problem: Calculate the probability of two people meeting at the same location between 1 and 2 p.m. Assume both people show and person 1 will wait 15 minutes for person 2.

Doesn't this probability increase as the time frame shrinks? For example, Isn't there a much smaller chance the pair will meet if person 1 arrives at 1:00 vs arrives at 1:45 (the probability of meeting becomes 1).

Is this somewhat easily answered using basic statistics? Just curious if my thought process is accurate or way wrong.

• Are we assuming a uniform distribution that person 2 shows up between 1pm and 2pm? similarly, are we assuming a uniform distribution that person 1 shows up between 1pm and 1:45pm? – Ian Coley Apr 12 '13 at 3:43
• I believe so. The original problem didn't address that, but I based my logic on the assumption that if this occurred 'naturally' (in a sterile environment without distractions, barring the preferences for time etc.) it would be a normally a distributed probability. – injuryprone Apr 12 '13 at 3:57

That's an example for the case where person1 will wait 10 minutes = $\frac 1 6$ hour for person2. To find the needed probability you should just integrate the marginal density over the shaded area.