Gambler’s ruin

Player A starts with k chips and Player B with N − k. Each round: A wins one chip from B with probability p (and loses one with probability q = 1 − p). The game ends when one player has all N chips.

For a fair coin (p = 0.5), the probability A wins is simply proportional to their starting stake:

P (A wins) = \frac{k}{N}

For a biased coin ( $p + q = 1$ ):

P (A wins) = \frac{1 - ( q / p ) ^{k}}{1 - ( q / p ) ^{N}}

P(A wins) theory

50.0%

simulated win rate

53.3%

color key

green = A wins · red = A ruined · grey = ongoing

A's starting chips (k) 10 Total chips (N) 20 p (A wins each round) 0.50 paths 30

A starts with k=10, B starts with 10. Each step: A gains 1 chip with prob p, loses 1 with prob 1−p.

P (A wins) = \frac{1 - ( q / p ) ^{k}}{1 - ( q / p ) ^{N}}

What to notice

Starting stake dominates. With a fair coin and A starting with 10 of 20 chips, A has a 50% chance. Move A to 5 of 20 chips — A’s chance drops to 25%. Small chip advantages translate linearly to win probability.
Even a tiny bias is catastrophic in the long run. Set p = 0.49 (A disadvantaged). As N grows, the formula shows A’s winning probability collapses exponentially — this is why casinos with small house edges reliably win over time.
Paths that don’t terminate (grey) have simply not reached either boundary within the step limit. Biased walks terminate faster; fair walks can meander for thousands of steps.

Why this matters

Gambler’s ruin is the canonical model for:

Bank solvency — a bank with finite capital facing random losses eventually fails, even with a slight edge.
Population genetics — a new gene variant competing against the existing population can drift to extinction even if slightly advantageous.
Insurance — an insurer with finite reserves facing random claims must eventually exhaust them without adequate reserves.

The lesson: finite capital + random walk = eventual ruin, unless the drift is strongly positive. Having more chips than your opponent matters as much as the coin itself.