A fair coin is flipped 100 times and a walk moves up or down 1 on each flip. Which is more likely, that the walk spends exactly 50 of the 100 steps on the positive side, or that it spends all 100 there? Give both probabilities.

A fair coin is flipped 100 times and a walk moves up or down 1 on each flip. Which is more likely, that the walk spends exactly 50 of the 100 steps on the positive side, or that it spends all 100 there? Give both probabilities.

Approach: Use the discrete arcsine law, which factorises the probability that exactly 2k of 2n steps are spent positive into u_{2k} u_{2n-2k} with u_{2j} = C(2j,j)/4^j, then evaluate both cases.

Spending all 100 steps positive is more likely, by a factor of 6.31. The arcsine law gives P(2k steps positive) = u_{2k} u_{2n-2k} with u_{2j} = C(2j,j)/4^j. For 2n = 100 the all-positive case is u_100 u_0 = C(100,50)/2^100 = 0.0796, while the even split is u_50 u_50 = (C(50,25)/2^50)^2 = 0.0126. The limiting density of the fraction of time spent positive is 1/(pi sqrt(x(1-x))), which is minimised at x = 1/2 and diverges at both ends, so the least likely outcome for a fair game is a shared random walk lead. Time in front is therefore a poor estimator of skill in a symmetric game.

Follow-up: What is the probability that the last visit to zero happens within the first ten steps?

Key concepts: arcsine law, time spent positive, limiting density, random walk lead.