# 2018-18 A random walk on the clock

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?

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# 2018-10 Probability of making an acute triangle

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.

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# 2014-24 Random points on a sphere

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!)

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# 2009-20 Expectation

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 $$\displaystyle \lim_{n\to \infty} e_n=\frac1e$$.

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