Airport shuttle and Jimmy > The airport shuttle arrives at the hotel at a random time between 7.30 am and 7.45 am. Jimmy waits at the hotel a random time between 7.30 am and 7.45 am (independently of the shuttle) and he will wait for (at most) $5$ min before leaving. Find the probability that Jimmy will catch the bus. So far the way I've thought about this problem is: $X \sim U[0, 15]$ where $X$ is the probability that the shuttle arrives at a specific minute. $Y \sim U[0, 15]$ where Y is the probability that Jimmy arrives at a specific minute. Find $P(Y\le X\le Y+15).$ Does this seem fine? I'm not sure entirely sure.

I managed to solve it by plotting a graph and then calculating the needed area.

To find the area I solved the following integral:

$\int_{0}^{15} \int_{0}^{x} \frac{1}{225}dy \space dx - \int_{5}^{15} \int_{0}^{x-5} \frac{1}{225}dy \space dx $

Which gives an answer of $\frac{5}{18}$