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jmc

counting and probability senior

Problem

Four distinct points, , , , and , are to be selected from points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord intersects the chord ?
Solution
Because all quadruples are equally likely, we need only examine the six clockwise orderings of the points: Only the first two of these equally likely orderings satisfy the intersection condition, so the probability is .
Final answer
\frac{1}{3}