Yeah, I saw that after I posted and had to think about how to dismiss p=1.


There cannot be a probability of 1 for going broke after N flips for any N no matter how large. This is because there is always a non-zero probability that we do not go broke after any N flips. For example, it is always possible to win all N times with probability (2/3)^N. In fact, as long as our number of losses does not exceed our number of wins by 3, we will not go broke, and this will have a non-zero probability for any number of flips. So we can't have p=1 or else the probability of going broke plus not going broke would be greater than 1 which is nonsense. It would also imply that our opponent with a negative expectation would have a positive expectation of 3 dollars after N flips which we know can't be right.


So you can make as many flips as you please, and the probability of losing our last dollar will never exceed 1/2, and the probability of losing 3 dollars is 1/8.