Hello all. I recently calculated the chances of doubling 1000$ using Martingale method. I did this with two methods: 1) Analytical 2) Numerical

First some definitions:

Roll: a single roll, you either win or lose, chances are 50% win and 50% lose.

5R method: in this method gambler believes he can't lose 5 times in a row, so he bets (amount he has)/((2^5)-1), that is (amount he has)/31.

Round: For 5R method round ends either when gambler wins before he runs out of money or loses all 5 rolls and has no more money to bet.

1) Analytical: In this scenario the gambler believes that there is no way he is going to lose 5 times in a row, chooses 5R method, so he starts with 1000$ and bets (1000/31)$, and each time he loses, he bets double previous amount (2000/31)$. I calculated that to double 1000$ (win 1000$) he must complete 22 rounds with wins. Now, the chances that he is going to lose 5 times in a row and leave the casino bankrupt are (1/2)^5, that is 1/32. So the chances he is going to go bankrupt before he completes 22 rounds and doubles his money are 22/32, that is 68.75%.

Result: 68.75% that he will lose 1000$ he had at the beginning, 31.25% that he will win 1000$

2) Numerical method: I used turbo pascal and wrote a simulation for this scenario, will not in detail.

Result: 50% he loses 1000$ he had, 50% he wins 1000 and leaves casino with 2000$ in his pocket.

Question: I really think that I must have made a mistake in my analytical calculation. But I check it over and over and can't find what is wrong with it. Can you guys help me? If you have any clarification please tell me.