Two segments inside a square.

In a square ABCD, points P and Q are chosen randomly and independently, find the probability that the SEGMENTS AP and BQ intersect.

Solution:

\(\dfrac{1}{4}\)

Idea:

1. Consider the convex hull of \(APBQ\). The probability that it forms a quadrilateral is \(\dfrac{1}{2}\).

Why?

Let P be the point closer to side CD, then Q must lie outside the triangle APB.

2. If \(APBQ\) forms a convex quadrilateral, we can always swap \(P\) and \(Q\).