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.

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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.

I am sure you can easily see any answers on your questions on this geometrical example.

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