medium Probability

Convex Quadrilateral

A release-checked medium problem for training Combinations and Permutations, Events, Continuous Random Variables.

Question

Suppose that $10$ points are randomly placed on the edge of a circle, and you draw all possible chords between the points. If you uniformly and randomly select $4$ chords, what is the probability that these $4$ chords form a convex quadrilateral? A convex quadrilateral is any four-sided polygon where every interior angle is less than $180$ degrees.

Practice focus

This Probability problem is tagged Combinations and Permutations, Events, Continuous Random Variables. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.