# What is wrong with my method for probability that n points on a circle are in one semicircle

So I understand the method used in this solution, and I know my method is incorrect, but I was just looking for an explanation why.

I was thinking that if I choose any spot on the circumference, there's a $$1/2^{n}$$ probability that all $$n$$ points lie in the semicircle starting from that spot on the circumference. Why couldn't you just integrate this probability from 0 to 1 representing moving the spot around the whole circle?

Again, I know this is the wrong and just looking for reasoning why.

E.g. let $$E_\theta$$ be the event that all the $$n$$ points are in the semicircle starting at the angle $$\theta$$ and going clockwise. Then for all $$\theta, P(E_\theta) = 1/2^n$$. But how do you add $$E_\theta$$ and $$E_{\theta + \epsilon}$$ for some small $$\epsilon$$? Those are overlapping semicircles. I find it hard to imagine any integral like $$\int_0^{2\pi} P(E_\theta) \,\,\text{blah} \,\, d \theta$$ would give you the right answer.
In particular, your original suggestion is equivalent to saying $$\text{blah} = {1\over 2\pi}$$, which is the density for picking the semicircle uniformly, so the result is $$\int_0^{2\pi} P(E_\theta) {1\over 2\pi} d \theta = 1/2^n =$$ Prob that a random semicircle (the one at a random $$\theta$$) contains all $$n$$ points (as Brian also pointed out).
Your reasoning would be correct if you wanted the probability that all $$n$$ points are in a random semicircle. The desired probability, however, is the probability that all $$n$$ points all fall within the same (but not predetermined) semicircle.
Thus, for example, a single point obviously falls within the same semicircle as itself. However, integration gives the probability that the point falls within a randomly selected semicircle, which is $$1/2$$.
• So I understand the probability all $n$ points are in a random semicircle is $1/2^{n}$, and I know integrating from 0 to 1 doesn't change this. But is there a formulation or way to integrate this probability moving the beginning of the semicircle all the way around the circle? – jtanman Mar 25 at 23:31