Please show all detailed steps in solution. The binary model system is mathemati

Question

Please show all detailed steps in solution. The binary model system is mathematically identical to the random-walk problem in one dimension. The random walk can be visualized as follows. A drunk starts from a lamp post and takes steps of length ` to the left or to the right at random with equal probability, using the curb as a guide, so that all his steps fall on a line. What is the probability that after N steps, he will (a) be back at the lamp post, or (b) be at a distance n` from it? Assume that N and n are even. Give approximate values of the two probabilities for N = 60, n = 20.

Explanation / Answer

a) P(at 0) = 30 steps to the left and 30 steps to the right = 60C30 * (1/2)^30*(1/2)^30 = 0.10257

b)P(at distance = 20 on either side) = P(40 steps to right and 20 steps left) +P(40 steps to left and 20 steps right)= 2 * P(40 steps to right and 20 steps left)=2*60C40 *(1/2)^60= 2* 60!/(40!*20!)*1/2^60=0.0072

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