Two random subsets of five elements overlapping

Two subsets $A$ and $B$ of $\{1,2,3,4,5\}$ are chosen independently and uniformly from all $32$ subsets. What is the probability that they share at least one element?

$781/1024$. Each element is independently in both sets with probability $\tfrac14$. The sets are disjoint when no element is in both, which has probability $(3/4)^5=243/1024$, so the answer is $1-\tfrac{243}{1024}$.