A man and a woman agree to meet at a certain location about 12:30 p. m. if the man arrives at a time uniformly distributed between 12:15 and 12:45 and the woman independently arrives at a time uniformly distributed between 12:30 and 1 pm, find the probability that the man arrives first

We define the spaces of the man and the woman's independent arrivals.

In this case,

f (man) = 1 (45-15) = 1/30

f (woman) = 1 / (60-0) = 1/60

f (man) * f (woman) = 1/1800

Probability = P

P (man-woman) < 5 = 1/1800 * integral of (y) with limits of (x+5) and (x-5) from 45 to 15.

P (man-woman) <5 = 1/6