Expected winnings in betting game

Question

Suppose you are playing a game where you are betting dollars and if you flip a coin and it is heads, then you win that amount, but if it's tails, you lose that amount. You use the strategy that you start by betting $\$1$. If you lose, you bet $\$2$. If you lose, you bet $\$4$ dollars. So in round $n$, you are betting $2^{n-1}$ dollars. However, if you win, you stop playing.

What is the net winning after we win on the $n$th coin flip? How do we find the expected value of our winnings in the $i$th round and our expected net winnings after the $n$th round?

Answer 1

Suppose that you win on the $n$th turn, then your net gain is $$2^{n-1}-(1+2+\dots+2^{n-2})=2^{n-1}-(2^{n-1}-1)=1$$

This betting strategy is called a martingale, and is the origin of the term martingale in probability theory. The downside of course is that the game can last arbitrarily long, so you need an unlimited amount of money to employ it.