Expected value and Gambler's fallacy Betting on a fair coin has expected value 0 dollars.
Suppose we win 1 dollar for each win and lose the same for each loss. Suppose we have lost 100 dollars so far. Then it's right to say that this loss has to be balanced out by the winnings somewhere in the future tosses of the coin? That's because the expected value is 0, so we can't remain at -100 dollars till infinity. But that also implies that the set of future tosses of the coin are overall biased towards winning, which is Gambler's fallacy. Please help.

You’re mixing up various things here. The fact that the expected value is zero doesn’t imply that we end up at any particular point; it’s just an expected value. To disentangle the various facts involved, imagine that you only bet $2^{-n}$ dollars on the $n$-th bet. Then your total profit will never be $0$ after the first bet; it will always be positive if you win the first bet and always negative if you lose the first bet. Yet the expected value of each bet and of the sum of the bets is zero.
Nevertheless, it so happens that if you do always bet the same amount, you will return to a net profit of $0$ at some point with probability $1$. As the above example shows, this does not follow from the fact that the expected value is $0$. It’s a further fact that needs to be proved separately. It doesn’t imply that the future tosses are biased; in fact it holds for arbitrary amounts: Given any (not necessarily positive) integer profit $a$ you have obtained after some number of bets and any (not necessarily positive) integer target $b$, the probability that you will at some point have a profit of $b$ is $1$. So if you’re at a profit of $-100$, not only will you almost surely return to a profit of $0$, you will also almost surely reach a profit of $-200$, which shows that no bias is implied.
