For the symmetric one-dimensional random walk (steps with probability each)

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods

Error, Chance and the Machine, Ch Random walks and Markov chains

Problem

For the symmetric one-dimensional random walk (steps ±1\pm 1 with probability 12\tfrac12 each), the probability that the first return to the origin occurs exactly at step 2n2n is f2n=12n−1(2nn)4−n.f_{2n} = \frac{1}{2n-1}\binom{2n}{n}4^{-n}. Compute f6f_6, the probability that the first return to the origin happens exactly at step 6.

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