Maryanne's mail arrives at a random time between 1 p.m. and 3 p.m. Maryanne chooses a random time between 2 p.m and 3 p.m. to go check her mail. What is the probability that Maryanne's mail has been delivered when she goes to check on it?
I solved it geometrically constructing a cartesian plane with the restrictions of the problem and imposing that Maryanne's checking of the mail was after the delivering of it. The solution with this approach is 3/4. However I'm trying to solve this problem in another way without using a plane where to count dimensions of areas. But I'm a bit stuck, Which can be an idea?
Let $X$ be the time that the mail arrive and let $Y$ be the time that Maryanne check her mail.
$X \sim Uni(1,3)$ and $Y \sim Uni(2,3)$,
\begin{align} P(X \le Y) &= \int_1^3 P(X \le Y|X=x) f_X(x) \, dx \\ &= \int_1^2 P(X \le Y|X=x) f_X(x) \, dx + \int_2^3 P(X \le Y|X=x) f_X(x) \, dx \\ &= \frac 12\int_1^2 P(X \le Y|X=x) \, dx + \frac12 \int_2^3 P(X \le Y|X=x) \, dx \\ &=\frac12 + \frac12 \int_2^3 (3-x) \, dx \end{align}
Use a substitution and you should be able to find the solution.