Suppose I'm gambling using a strategy of doubling my bet whenever I lose to recoup my losses from previous bets. If my initial bet is 1/2048 of my capital, I can bet 10 times before I run out of money. Statistics says that the chances of this happening are ~1/718. As the law of large numbers states, though, this is in no way a guarantee that such an outcome will manifest in any certain number of iterations.
Is it possible, therefore, to calculate the chances that this doesn't happen? For example, is there an equation we can put x (the number of bets) into and determine the likelihood of this 1/718 chance not occurring? My chances of losing within the first ten bets would be extremely small, and my chances after ten thousand would be pretty high, so is there any way to calculate the middle ground?