A man and a woman decide to meet at a certain location. If each of them independently arrives at a time uniformly distributed between 12 noon and 1 P.M., find the probability that the first to arrive has to wait longer than 10 minutes.

Let X and Y denote, respectively, the time past 12 that the man and the woman arrive, then X and Y are independently random variables, each of which is uniformly distributed over (0, 60). The desired probability,

P[X + 10 < Y] + P[Y + 10 < X]= 2P[X + 10 < Y]

But why

and not

in the first part of the integral??