Question: 1. 2.37 Aparticle movesoverthe flat surface of a grid such that an equal unit of distance is measured withevery step. The particle begins at the

1. 2.37 Aparticle movesoverthe flat surface of a grid such that an equal unit of distance is measured withevery step. The particle begins at the origin (0,0). The first step may be to the left, right, up or down, with equal probability 1 4

. The particle cannot move back in the direction that the previous step originated from. Each of the remaining three directions has an equal probability of 1 3

. Suppose that the particle makes a total of n steps for a given value of n. Verify experimentally that the expected value of the distance between the particle’s starting and ending points is approximately equal to 1.25√

n if n is sufficiently large. Also, for n = 25 and n = 100, find the probability that the maximal distance of the particle to its starting point during the n steps will exceed 1.65√

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