Question

2. We distribute n points uniformly and independently on the circumference of a circle, and want to compute the probability that there is a semicircle that contain all of them. (In other words, the probability that there is a line through the center of the circle such that all n points lie on the same side of this line.) Let E be the event that such a semicircle exists. Denote by Pi, P2, ..., Pn the random points, and by Ei the event that each point is contained in the semicircle that starts from P in the counterclockwise directioni 1,2,..., n) (a) (1 point) Write E in terms of E (b) (1 point) Do we have P(Ei n E0 for all i j? (c) (2 points) Calculate P(E). (d) (2 points) Now answer this question: if we drop n points uniformly and i dependently on a disk, what is the probability that the center of the disk is contained in the set formed by convex combinations of the n points? (That is, inside the convex polygon of the random points as vertices.) Mind the previous version with the circumference of the circle.

Answer 1

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