48. An immortal drunk man wanders around randomly on the integers. He starts at the

Chapter 2, Problem 48

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48. An immortal drunk man wanders around randomly on the integers. He starts at the origin, and at each step he moves 1 unit to the right or 1 unit to the left, with probabilities p and q = 1p respectively, independently of all his previous steps. Let Sn be his position after n steps. (a) Let pk be the probability that the drunk ever reaches the value k, for all k 0. Write down a dierence equation for pk (you do not need to solve it for this part). (b) Find pk, fully simplified; be sure to consider all 3 cases: p < 1/2, p = 1/2, and p > 1/2. Feel free to assume that if A1, A2,... are events with Aj Aj+1 for all j, then P(An) ! P([1j=1Aj ) as n ! 1 (because it is true; this is known as continuity of probability).