Play a fair game long enough against a bigger bankroll and you WILL go broke — not because the game is unfair, but because you have less to lose. With even a tiny house edge, ruin becomes near-certain. It’s the cold arithmetic under ‘the house always wins’ and why bankroll, not just edge, decides survival. Slide the edge and watch the ruin probability climb.
The Gambler’s Ruin: staking 1 unit per round on a bet won with probability p (q=1−p) until you reach N or hit 0, the probability of RUIN from a stack of i is (for p≠q) [(q/p)^i − (q/p)^N]/[1 − (q/p)^N], and simply 1 − i/N for a fair game p=0.5. Even a fair game ruins the smaller bankroll in proportion to the size gap; any negative edge makes ruin approach certainty as play continues. A fail-loud self-check throws unless a fair game from 5 of 10 gives 50% ruin and a losing edge gives more. ◆ real probability, node-verified.
Exact for fixed unit bets and these boundaries; bet-sizing (see [[the-kelly-criterion]]), table limits, and quitting rules change the picture. The lesson — bankroll size governs survival as much as edge — is the durable one.