Suppose that n points are independently chosen atrandom on

Chapter 6, Problem 6.16

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Suppose that n points are independently chosen atrandom on the circumference of a circle, and wewant the probability that they all lie in some semicircle.That is, we want the probability that there isa line passing through the center of the circle suchthat all the points aLet P1, . . . ,Pn denote the n points. Let A denotethe event that all the points are contained in somesemicircle, and let Ai be the event that all thepoints lie in the semicircle beginning at the pointPi 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).re on one side of that line, asshown in the following diagram:Let P1, . . . ,Pn denote the n points. Let A denotethe event that all the points are contained in somesemicircle, and let Ai be the event that all thepoints lie in the semicircle beginning at the pointPi 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).