The classic setup
The gambler’s ruin problem is one of the oldest results in probability. A player starts with a fixed number of units and repeatedly makes a one-unit bet. Each bet is won or lost independently. The play stops when the player reaches a chosen target or when the money hits zero. The question is: what is the chance of reaching zero first?
It is a simple model of a finite bankroll, and the answer is more sobering than most people expect.
It is a random walk with two walls
Picture the bankroll as a marker on a number line. Each win moves it one step up and each loss moves it one step down. Zero is a wall that ends the game. Mathematically this is a random walk with absorbing boundaries. Because the walk can wander, any finite budget has some chance of touching zero.
A fair game
Take a fair game, with a 50% chance of winning each bet. A player starting with a units and playing until reaching N units or zero has a chance of ruin of 1 - a/N.
Small example, an illustration: start with 1 unit and aim for 5 units. The chance of ruin is 1 - 1/5 = 0.8, and the chance of reaching the target is 0.2. Start with 2 units aiming for 4: the chance of ruin is 1 - 2/4 = 0.5.
Even in a perfectly fair game, a player who keeps going until a target or zero will be ruined more often when the target is large compared with the starting amount. If the target is ever-higher, or there is no target at all, the chance of eventual ruin approaches certainty, because the walk is free to wander all the way down given enough time.
A game with a negative expectation
Now make the game unfavourable. Suppose each bet is won with probability 0.4 and lost with probability 0.6. This is a deliberately large edge, chosen so that the arithmetic stays visible. Start with 1 unit and aim for 3.
Let h be the chance of reaching 3 from a given amount. From 2 units, a win (0.4) reaches the target and a loss (0.6) drops to 1. From 1 unit, a win (0.4) moves to 2 and a loss ends the game. That gives:
- h(1) = 0.4 x h(2)
- h(2) = 0.4 + 0.6 x h(1)
Substituting, h(2) = 0.4 + 0.6 x 0.4 x h(2) = 0.4 + 0.24 x h(2), so h(2) = 0.4 / 0.76, about 0.526. Then h(1) = 0.4 x 0.526, about 0.21. The chance of ruin is about 0.79, noticeably worse than the 0.67 you would get in a fair game with the same start and target (1 - 1/3).
The same calculation extends to longer ladders. When the odds are unfavourable, the extra edge compounds against the player at every step.
What “eventually runs out on average” means
For games with a negative expected value, the long-run picture is clear. Each bet costs the player a little on average. A game with a house edge, such as an even-money bet on a single-zero roulette wheel, has an expected result of about -1/37 per unit staked. With no target and no stopping rule, the probability of eventually reaching zero with a finite bankroll is 1. The careful way to say this is that ruin becomes certain given unlimited play, and that the average outcome is a loss that grows with the total amount staked.
It does not mean every session ends badly. Short runs vary widely, and many sessions end ahead. But when the same finite bankroll is exposed to a negative edge repeatedly, the possible gains are capped by when the player chooses to stop, while the losses are capped only by the money available.
Common misreadings
- Ruin does not take long to arrive in a specific session; it is a long-run statement about probability.
- Larger starting money lowers the chance of ruin before a given target, but it does not change the sign of the edge. Against a negative edge, a larger bankroll only delays the typical result and increases the expected total loss as more is staked.
- Raising the stake after losses, or chasing losses, is not a counter to ruin. It speeds up the walk towards the boundary, since fewer steps remain.
- Past losses do not make a win more likely. That belief is the gambler’s fallacy.
Why the model matters
The problem explains why the house, which has a large bankroll and a built-in edge, does well over many plays, while an individual with a limited budget faces a real chance of losing it all. It is a clear reason to treat any spending on gambling as a cost rather than an income.
This is also the reasoning behind simple ideas such as setting a firm limit before playing. Never borrow to play, and do not stake more than you can afford to lose. If gambling is affecting your finances or wellbeing, contact a local support service.