Suppose that we are given 12 points evenly spaced on a circle. Starting from a point in the 12 o’clock position, a particle P will move to one of the adjacent positions with equal probably, 1/2. P stops if it visits all 12 points. What is the most likely point that P stops for the last?
Given a stick of length 1, we choose two points at random and break it into three pieces. Compute the probability that these three pieces form an acute triangle.
Suppose that \(n\) points are chosen randomly on a sphere. What is the probability that all points are on some hemisphere?
(This is the last problem of 2014. Thank you!)
Let en be the expect value of the product x1x2 …xn where x1 is chosen uniformly at random in (0,1) and xk is chosen uniformly at random in (xk-1,1) for k=2,3,…,n. Prove that \(\displaystyle \lim_{n\to \infty} e_n=\frac1e\).
If we choose n points on a circle randomly and independently with uniform distribution, what is the probability that the center of the circle is contained in the interior of the convex hull of these n points?