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# Bankrupt

Two players,

What is the probability that player

*A*and*B*, play a game in which the winner receives 1 dollar from the other player. Player*A*has 1 dollar and player*B*has 2 dollars. Player*A*is better in this game and wins 2/3 of the games. They play until one of them is bankrupt.What is the probability that player

*A*wins?
Hint

Take a look at the lesson about Markov Chain Probability (Probability).

Solution

This is a pretty straightforwards Markov chain problem. There are 4 states. The transition graph is given in Figure 1.

The problem starts at state 1. As has been explained in the lessons of this course, we use the following equation:

If we substitute Equation 4 in Equation 3, we have

The problem starts at state 1. As has been explained in the lessons of this course, we use the following equation:

(1)

(2)

Furthermore, and . Then we have(3)

(4)

Solving these equation gives us and . So, starting with 1 dollar, player*A*has a 4/7 chance of winning.**Proof**If we substitute Equation 4 in Equation 3, we have

(5)

(6)

(7)

(8)

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