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This question is really throwing me off:

Lets say there's two players, A and B. Each game consists of betting \$1. Gameplay ends when one player has all of the money. Player A starts with \$3, B starts with $5. If Player A has probability of winning 2/3, what's the probability that Player A will win the whole thing?

I understand that we want to take into account the probability that player A will win 5 in a row: $(2/3)^5$ and add it to the probability that after some games we'll be back at the start. Not sure where to go forward though.

Grant Seltzer
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1 Answers1

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For $k\in\left\{ 0,1,\dots,8\right\} $ let $p_{k}$ denote the probability that $A$ will win the whole thing if $A$ starts with $k$ dollars.

Then $p_{0}=0$, $p_{8}=1$ and $p_{k}=\frac{2}{3}p_{k+1}+\frac{1}{3}p_{k-1}$ for $1\leq k\leq7$.

Do you understand why?

Now find a solution for these equations. You are actually looking for $p_3$.

drhab
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  • I believe I understand why, but how would you ever be able to evaluate that? – Grant Seltzer Apr 30 '16 at 18:50
  • As I said: by finding a solution for the mentioned equations. – drhab Apr 30 '16 at 18:51
  • So p_0 = 0, p_1 =2/3p_2 + 0, p_2 = 2/3p_3 + 1/3(2/3p_2) and so on? Couldn't that never come to on conclusive answer since all values p_k for 0<= k <= 8 are dependent on one another? – Grant Seltzer Apr 30 '16 at 18:57
  • With that you are on the right track. At first you can find an expression for $p_2$ in $p_1$. Then you can find an expression for $p_3$ in $p_1$. And so on. Finally you find an expression for $p_8$ in $p_1$. This expression in $p_1$ must equalize $1$ which gives you the opportunity to determine $p_1$. After that you can easily determine the others. There is only one thing to do: trying it out. – drhab Apr 30 '16 at 19:02
  • I don't want seem unappreciative or lazy, but I can't figure this out.

    I wrote out $p_3$ in terms of all the other $p_k$ and don't know how this could evaluate.

     – Grant Seltzer Apr 30 '16 at 19:24
  • Can you find an expression for $p_2$ in $p_1$? If so then can you find an expression for $p_3$ in $p_1$? If so then where does it stop? – drhab Apr 30 '16 at 19:34
  • by $p_2$ in $p_1$ do you mean $p_1$ = $2/3$$p_2 + 0?$ – Grant Seltzer Apr 30 '16 at 19:37
  • Let us continue this discussion in chat. – drhab Apr 30 '16 at 19:37