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Given $n$ random points on a circumference, what is the probability of them lying in the same semicircumference?

Feedback on my solution and/or alternative solutions would be appreciated.

EDIT This is a duplicate of math.stackexchange.com/q/18369/27978

Ignacio
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1 Answers1

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Given $n$ points, build $n$ points opposite to the initial ones. Then either the initial point or its opposite lies in any given semicircumference.

Fix 2 initial points (they necessarily lie in the same circumference). For each of the remaining $n-2$ points, the probability of lying in the same semicircumference as the first 2 points is 0.5. The events are independent.

Hence the overall probability is $0.5^{n-2}$

Ignacio
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  • The $2$ initial points do not determine a semicircumference. If the initial points are very close to eachother then the probability that a semicircumference exist that contains the first $2$ points and also the third point is not $0.5$ but almost $1$. – drhab May 23 '18 at 17:01
  • But points are random - that means we have to factor in the probability of these first 2 being very close? – Ignacio May 23 '18 at 17:05
  • I only took them "close" to make extra clear that your solution is not okay (almost $1$ instead of $0.5$). – drhab May 23 '18 at 17:08