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Bankrupt
Two players, 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?
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.
Figure 1 - Transition graph for this problem.
The problem starts at state 1. As has been explained in the lessons of this course, we use the following equation:
and 
. Then we have
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
The problem starts at state 1. As has been explained in the lessons of this course, we use the following equation:
(1) 
(2) 
and . Then we have
(3) 
(4) 
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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