Canada Math Contest #2 - Problem 9

Full credit goes to the staff team at Canada Math for authoring these problems.

Jonas has a square canvas $ABCD$ of side length 1. He draws two points $E$ and $F$ independently and at random such that $AE = BF = 1$. If the probability that segments $AE$ and $BF$ intersect is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, find $p+q$.