probability of 2 segments from 10 points on a circle intersect

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Source: OLCOMA Costa Rica National Olympiad, Final Round, 2018 3.1

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#1 h Sep 20, 2021, 5:06 PM

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There are $10$ points on a circle and all possible segments are drawn on the which two of these points are the endpoints. Determine the probability that selecting two segments randomly, they intersect at some point (it could be on the circumference).

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#2 h Sep 21, 2021, 8:49 AM

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We have $\binom{\binom{10}{2}}{2}=990$ options to choose two segments.
Now, let's see how many of these pairs intersect at some point. We have $2$ different situations.
Number of pairs intersecting on the circumference:
We have $10$ options to choose the endpoint that the pair intersect. So, we have $9$ points left and we should choose two of them.
Hence, we have $10\cdot \dbinom{9}{2}=360$ pairs of segments that intersect on the circumference.
Number of pairs intersecting inside the circle:
Each $4$ different points form exactly $1$ pair of segments that intersect inside the circle. Thus, we have $\dbinom{10}{4}=210$ such pairs.

In total, we have $360+210=570$ pairs that intersect at some point. The total number of pairs is $990$ .
Hence, the probability is $\dfrac{570}{990}=\frac{19}{33}$ .

This post has been edited 1 time. Last edited by BarisKoyuncu, Sep 21, 2021, 8:50 AM
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#3 h Sep 21, 2021, 12:36 PM

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If we choose four points on the circumference, there are exactly one form two draw 2 segments that cut internally and the other two forms to draw 2 segments doesn't cut. Also we count each pair of segments once. We conclude that the probability is 1/3

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