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: $\sum_{n= 100}^\infty \begin{pmatrix}n \\ \frac{n+ 200}{2}\end{pmatrix}\left(\frac{4}{13}\right)^{\fr ac{n+ 200}{2}}\left(\frac{9}{13}\right)^{\frac{n- 200}{2}}$.