Coin flip gamble 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?
 A: 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: 


*

*Do you have to bet exactly $x$ times (i.e., you cannot bet all of your money until the final bet)?

*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.
A: 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.
