
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?

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= (n200)/2. The probability of winning 200+ k= 200+ (n 200)/2= (n+ 200)/2 times and losing (n 200)/2 times is $\displaystyle \begin{pmatrix}n \\ \frac{n+ 200}{2}\end{pmatrix}\left(\frac{4}{13}\right)^{\fr ac{n+ 200}{2}}\left(\frac{9}{13}\right)^{\frac{n 200}{2}}$. The probability of ever doubling your money is the sum of that over all n: $\displaystyle \sum_{n= 100}^\infty \begin{pmatrix}n \\ \frac{n+ 200}{2}\end{pmatrix}\left(\frac{4}{13}\right)^{\fr ac{n+ 200}{2}}\left(\frac{9}{13}\right)^{\frac{n 200}{2}}$.