Probability that n points lie on one side of a circle

[CAF123]

Homework Statement

Suppose that n points are independently chosen at random on the circumference of a circle and we want the probability that they all lie in some semicircle.
Let ##P_1...P_n## denote the n points. Let A denote the event that all the points are contained in some semicircle and let ##A_i## be the event that all the points lie in the semicircle beginning at the point ##P_i## and going clockwise for 180 degrees, ## i \in {1,...n}## Find P(A).

The Attempt at a Solution

I know $$ P(A) = P(\cup_{i=1}^{n} A_i) = \sum_{i=1}^{n} P(A_i) $$ since the ##A_i## are mutually exclusive. I then said each ##P_i## is a uniformly distributed point with density $$f(x) = 1/(2\pi), x \in (0,2\pi). $$
So $$P(0≤x≤\pi) = \int_{0}^{\pi} 1/(2\pi) dx = 1/2. $$ So this is the probability of one point being in a semicircle. (this result being obvious) So for the first point, I have $$P(A_1) = 1/2 (1/2)^{n-1}. $$ Similarly, $$P(A_2) = (1-1/2)(1/2)(1/2)^{n-2} $$ and $$P(A_3) = (1-1/2)(1-1/2)(1/2)(1/2)^{n-3} ...$$

Bringing this together, I get $$P(A) = P(A_1) +P(A_2) +...P(A_n) = n(1/2)^n, $$ after simplification. But the answer says ##n(1/2)^{n-1}?## Any advice? Many thanks
EDIT: I thought it through a little more. The probability that the first point starts the semicircle is 1/n. The probability of the remaining n-1 points lying in the semicircle is ##(1/2)^{n-1}## Would ##P(A_1) = 1/n (1/2)^{n-1}?##
EDIT2: Sorry, misread the Q. ##P(A_1)## is the probability that all the points lie in a semicircle beginning at point 1. This is a given, but I previously assumed they wanted to know the probability that ##P_1## was the first point. Hence, fixing point 1, the probability that all the remaining points lie in the semicircle going clockwise is 1/2 for each point (either it is in a semicircle going clockwise or anticlockwise) So ##P(A_i) = (1/2)^{n-1} => P(A) = n (1/2)^{n-1}.## Is this now a reasonable argument?

 

[haruspex]

Looks like you got there. The argument is sound.

 

[Ray Vickson]

Homework Statement

Suppose that n points are independently chosen at random on the circumference of a circle and we want the probability that they all lie in some semicircle.
Let ##P_1...P_n## denote the n points. Let A denote the event that all the points are contained in some semicircle and let ##A_i## be the event that all the points lie in the semicircle beginning at the point ##P_i## and going clockwise for 180 degrees, ## i \in {1,...n}## Find P(A).

The Attempt at a Solution

I know $$ P(A) = P(\cup_{i=1}^{n} A_i) = \sum_{i=1}^{n} P(A_i) $$ since the ##A_i## are mutually exclusive. I then said each ##P_i## is a uniformly distributed point with density $$f(x) = 1/(2\pi), x \in (0,2\pi). $$
So $$P(0≤x≤\pi) = \int_{0}^{\pi} 1/(2\pi) dx = 1/2. $$ So this is the probability of one point being in a semicircle. (this result being obvious) So for the first point, I have $$P(A_1) = 1/2 (1/2)^{n-1}. $$ Similarly, $$P(A_2) = (1-1/2)(1/2)(1/2)^{n-2} $$ and $$P(A_3) = (1-1/2)(1-1/2)(1/2)(1/2)^{n-3} ...$$

Bringing this together, I get $$P(A) = P(A_1) +P(A_2) +...P(A_n) = n(1/2)^n, $$ after simplification. But the answer says ##n(1/2)^{n-1}?## Any advice? Many thanks
EDIT: I thought it through a little more. The probability that the first point starts the semicircle is 1/n. The probability of the remaining n-1 points lying in the semicircle is ##(1/2)^{n-1}## Would ##P(A_1) = 1/n (1/2)^{n-1}?##
EDIT2: Sorry, misread the Q. ##P(A_1)## is the probability that all the points lie in a semicircle beginning at point 1. This is a given, but I previously assumed they wanted to know the probability that ##P_1## was the first point. Hence, fixing point 1, the probability that all the remaining points lie in the semicircle going clockwise is 1/2 for each point (either it is in a semicircle going clockwise or anticlockwise) So ##P(A_i) = (1/2)^{n-1} => P(A) = n (1/2)^{n-1}.## Is this now a reasonable argument?

If you use the results of problem 18, page 41 of Feller, "An Introduction to Probability Theory and its Applications", Vol. II (Wiley, 1971), the answer will come out as
[tex] P_{Feller} = \frac{n+1}{2^n}, \: n \geq 3 .[/tex] For n = 3 this gives a probability of P_Feller = 1/2, while the answer you wrote gives 3/4. (Note: the n in Feller is the same as the n in your problem!)

RGV

 

[haruspex]

For n=3, I believe 3/4 is correct. We can fix one of the points as an origin and measure the locations of the other two from there as being in the range [0, 1], uniformly distributed. We can represent the pair graphically as (x, y), the possibilities forming a unit square.
If x<0.5 and y<0.5, the condition is met.
If 0.5<x and 0.5<y, the condition is met.
If |x-y| > 0.5 the condition is met.
These are non-overlapping areas adding up to 3/4.