A game is played so that the wealth of the gambler at each play either rises by 1 with probability .51 or falls by 1 with probability .49 , until the wealth either hits 0 or 8 , at which point the game stops. For each interior state \(1,2, \ldots, 7\) in the transition diagram below, find the probability that it takes \(k\) units of time to reach state 8 , for values of \(k=1,2,3,4,5,6,7,8,9,10\). Comment on whether, if the computation were continued indefinitely, these probabilities would sum to 1 .

Transcribed Image Text:

0 0.49 0.49 0.49 0.49 0.49 0.49 0.51 0.51 0.51 0.51 0.51 0.51 0.49 1 2 3 5 6 7 0.5 8 Exercise 7