Coin flip gamble

Question

You have an amount of money to bet on a fair coin flipping and landing on heads. How much should you bet as a function of your balance to maximize your probability of profiting if you play $x$ times?

Answer 1

If it is a truly fair coin, with a 50% chance of each side coming up, the chance of winning and the chances of losing are equal, and losing $X$ dollars or gaining $X$ dollars cancel out to an average profit of $0$ dollars absolutely regardless of how much you bet.

Answer 2

Seems like the previous answers didn't fully appreciate the question, which I think has multiple interpretations.

Let $A$ be your initial amount of money, and $x$ be the number of times you need to bet. Now there are two ambiguities:

1. Do you have to bet exactly $x$ times (i.e., you cannot bet all of your money until the final bet)?
2. Do you have to bet the same amount each time?

Suppose (1) and (2) are both true. If $x$ is odd you have exactly a 50% chance of profiting, no matter what. However, if $x$ is even you have strictly less than a 50% chance of profiting. For example, if $x = 2$, then you have only a 25% chance of profiting.

Suppose (1) is false and (2) is true. Then clearly, you will simply bet some odd number of times $t \leq x$, and have a 50% chance of profiting.

Suppose (2) is false (regardless of (1)). Then you can actually achieve a $1 - .5^x$ chance of profiting. For each round $i$ until you win a coin toss, simply bet $A * .5^{x-i+1}$ dollars. $\sum_{i=1}^\infty A*.5^i = A$, so clearly you will never go broke, because $x$ is finite. However, each bet is more than the sum of all previous bets. Thus, once you win one coin toss, you will have more than $A$ dollars, and can simply bet a small enough sliver of your profits for the remaining bets to guarantee that you profit.