# Probability that $n$ random points lie on semicircle

If random points are chosen from a circle, what is the probability that all of them come from same semicircle ?

My reasoning is that; Let two random points be chosen first. Obviously then both of them lie in same semicircle. Now mark any diameter including those two on the same side of it. Any more point chosen will either lie within or beyond the semicircle made by diameter each with probability 1/2.

So 3 points lying on same semicircle is $\frac{1}{2}$.

4points lying in same semicircle is $\frac{1}{2}.\frac{1}{2} = \frac{1}{2^2}$
Similarly, for n such points the total probability is $\frac{1}{2^{n-2}}$.

Am I right in my reasoning?

## 1 Answer

No, your reasoning is not quite right. The first two points lie in many semicircles - in fact an infinite number. What proportion of all semi-circles do they lie in? That depends on how close they are to each other.

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