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Math Help - gamblers ruin

  1. #1
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    gamblers ruin

    hello does anyone know a simple way to find answers to questions like this? I have 100 chips , dealer has 200 chips maximum bet 1 chip . THE GAME . dealer has a standard pack of playing cards , and deals one card if its a JACK, QUEEN ,KING OR ACE, I WIN 1 chip any other card the dealer wins my 1 chip, after each game the card is replaced and the deck shuffled so I have 16/52 chance of winning each game we play . question 1 . what are my chances of doubling my money ? question 2 . what are my chances of losing all my chips before the dealer?
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  2. #2
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    Re: gamblers ruin

    Of the 52 cards in a deck, 16 are "jack, queen, king, or ace" and 36 are not. So, on any one play, your probability of winning is 16/52= 4/13 and your probabilty of losing is 36/52= 9/13. In order to double your money in n plays, you must win 200+ k times while losing only k times: 200+ k+ k= 200+ 2k= n so k= (n-200)/2. The probability of winning 200+ k= 200+ (n- 200)/2= (n+ 200)/2 times and losing (n- 200)/2 times is \begin{pmatrix}n \\ \frac{n+ 200}{2}\end{pmatrix}\left(\frac{4}{13}\right)^{\fr  ac{n+ 200}{2}}\left(\frac{9}{13}\right)^{\frac{n- 200}{2}}. The probability of ever doubling your money is the sum of that over all n: \sum_{n= 100}^\infty \begin{pmatrix}n \\ \frac{n+ 200}{2}\end{pmatrix}\left(\frac{4}{13}\right)^{\fr  ac{n+ 200}{2}}\left(\frac{9}{13}\right)^{\frac{n- 200}{2}}.
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