Two pairs of points are randomly chosen on a circle. Find the probability that the line joining the two points in one pair intersects that in the other pair.

Harrison Mills 2022-09-25 Answered
Two pairs of points are randomly chosen on a circle. Find the probability that the line joining the two points in one pair intersects that in the other pair.
I've been thinking over this problem, assuming one pair and finding that the other pair has to be entirely in one of the two arcs of the circle that the first pair of points divides it into.
But I've not been able to find an explicit answer. If I assume one point to be (a,b), I'm not able to manage the other three points.
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Answers (1)

xjiaminhoxy4
Answered 2022-09-26 Author has 9 answers
For any configuration of the four points, there are three ways they can be divided into pairs. That means that as you go through all possible placements of the two chords, each configuration of points, regardless of which are paiblack with which will appear three times. Only one of those ways will give intersection.

Extra intuition on why the answer is lower than 0.5: If you choose one pair first, and that pair's chord happens to be a diameter, then the probability that the other two points are on opposite sides of that diameter is 0.5. If the first chord is not a diameter, then the probability of intersection is less than 0.5.
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