Is this a sustainable betting strategy on a 50/50 game?

Question

There is a 50/50 chance of winning a game. I bet small on every game and wait for a certain number of losses in a row, say 5, and then bet big (more than 32 times the original bet) on the 6th game, thus increasing the probability of success. If I still continue to lose I then double up every time until I make back that 6th bet (this would be the Martingale system). I can't see the flaw but I feel like there is one? Can anyone give a probabilistic explanation as to why this strategy would fail?

Answer 1

There is no problem in what you describe. But your strategy assumes that you have an unlimited "bank" to bet from.

If you have a finite amount of cash in bank, then the probability of eventual bankrupt is equal to 1.

Answer 2

This is a perfectly sustainable strategy, if you can take out infinite loans. This is totally possible in a pure mathematical environment, and you can make a scenario in which you win infinitely; however, if you cannot take out loans, then the odds of your losing all of your money on $1$ run are $$\dfrac{1}{\log_2\left(\dfrac{\text{total money}}{\text{base bet}}\right)}$$ and your odds of doubling your money on $1$ run are once again $$\dfrac{1}{\log_2\left(\dfrac{\text{total money}}{\text{base bet}}\right)}$$ so you are just as likely to lose everything as to double your money on a run, so your expected payout per run is $\$0$. Please do not try this strategy in real life, because it will appear that you are steadily profitable until that awkward moment when you are not, and suddenly you have lost everything.