TITLE: Probability that the two segments intersect
QUESTION [5 upvotes]: P and Q are uniformly distributed in a square of side AB. What is the probability that segments AP and BQ  intersect?

REPLY [8 votes]: Inspired by Lee David Chung Lin's answer.

Consider the four events
$$AP \cap BQ \neq \varnothing, \\
AP \cap DQ \neq \varnothing, \\
CP \cap BQ \neq \varnothing, \\
CP \cap DQ \neq \varnothing.$$
Ignoring degenerate configurations, the events are mutually disjoint, equiprobable, and cover the whole space. $p = 1/4$.