Even in a fair game, the player with less capital eventually loses
Suppose you and a casino play a completely fair game in which you exchange money based on coin flips (with exactly a 50% chance of winning). But the casino has an unlimited amount of money, while you have a fixed amount of starting capital. If you keep playing until one of you goes broke, who will win?
Formula: If your capital is i and your opponent's capital is N-i, the exact probability that you win all the money (N) before your opponent goes broke is i/N. Conversely, the probability that you go broke first is (N-i)/N. When your opponent has much more capital (so N is larger), your probability of winning approaches 0 — even though the game is fair!
My capital10
Opponent's capital100
Probability that I win all of my opponent's money
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💡 This is one of the reasons casinos make money so consistently — a casino has much more capital than any individual customer, so even if each individual game is completely fair, the casino will almost always come out ahead in the long run.