Question

Problem 5: Gambler's Ruin Our old friend John Doe who tried his luck at blackjack back in Homework 2 now decides to win a small fortune using slot machines mstead. Having ganed some wisdom from his previous outings, he starts off small with just one dollar. He plays the slot machines in the following way He always inserts one dollar into the slot machines After playing it, the machine returns two dollars with probability p and returns nothing with probability 1-p John plays the machines until he has accumulated three dollars or has no money left Model the setting above as a DTMC by drawing the state space diagram. Represent Now try to find the probability of John losing all his money (also referred to as getting whichever one occurs first the transition probabilities between the states appropriately ruined) as a function of p To do this, define P as the probability of winning (getting three dollars) given that we start off with i dollars. We have Po0 and Ps 1. Now write P and P2 as a recursion and solve for them using the conditions for Po and P3. Finally conclude that the probability of getting ruined to be 1 - P1. Find the numerical value of this probability when p 0.6

Answer 1

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nber's forture at time n Evaluale both coses the Uc nad lo calciade P:o) for any n vave mutually exelgerts then After we neu ine msults por cock game the apilal of the A plages w be the inicial Coplal for new Gane, I hen ncreases or deteoses in one und. Thot op ued frm aK cap.tal P(w): P