A player plays a game in which, during each round, he has a probability 0.4 of w

Question

A player plays a game in which, during each round, he has a probability 0.4 of winning $1 and probability 0.6 of losing $1. These probabilities do not change from round to round, and the outcomes of rounds are independent. The game stops when either the player loses $2. its money or wins a fortune of SM. Assume M-4, and the player starts the game with (a) Model the player's wealth as a Markov chain and construct the probability transition matrix (b) What is the probability that the player goes broke after 4 rounds of play?

Explanation / Answer

state are 0,1,2,3,4

P= [1 0 0 0 0 ; 0.6 0 0.4 0 0 ; 0 0.6 0 0.4 0 ; 0 0 0.6 0 0.4 ; 0 0 0 0 1]

P =

    1.0000         0         0         0         0
    0.6000         0    0.4000         0         0
         0    0.6000         0    0.4000         0
         0         0    0.6000         0    0.4000
         0         0         0           0    1.0000

b)

P^4

ans =

    1.0000         0         0         0         0
    0.7440    0.1152         0    0.0768    0.0640
    0.5328         0    0.2304         0    0.2368
    0.2160    0.1728         0    0.1152    0.4960
         0         0         0         0    1.0000

since it is given that initial he has 2 $

hence we need to find P420

= 0.5328

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