# Question

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. That is, we want the probability that there is a line passing through the center of the circle such that all the points are on one side of that line, as shown in the following diagram:

Let P1, . . . ,Pn denote the n points. Let A denote the event that all the points are contained in some semicircle, and let Ai be the event that all the points lie in the semicircle beginning at the point Pi and going clockwise for 180˚, i = 1, . . . , n.

(a) Express A in terms of the Ai.

(b) Are the Ai mutually exclusive?

(c) Find P(A).

Let P1, . . . ,Pn denote the n points. Let A denote the event that all the points are contained in some semicircle, and let Ai be the event that all the points lie in the semicircle beginning at the point Pi and going clockwise for 180˚, i = 1, . . . , n.

(a) Express A in terms of the Ai.

(b) Are the Ai mutually exclusive?

(c) Find P(A).

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