Gambler's ruin explained

What gambler's ruin means

The Wikipedia entry on gambler's ruin (opens in a new tab) describes it as the idea that a gambler playing a game with negative expected value will eventually go bankrupt, regardless of their betting system. Every standard casino bet has a negative expected value for the player, because of the house edge. So in the long run, a player who keeps going has only one destination.

There is a second, less obvious version. Even in a completely fair game, with no edge at all, a player with a finite amount of money betting against an opponent with effectively unlimited wealth will eventually and inevitably go broke. The player can be wiped out; the opponent can't. A casino, backed by far more money than any one customer, is that opponent.

None of this says you will lose on a given night. It says what happens if you never stop, and it puts numbers on how likely you are to reach a goal before your money runs out.

Where the gambler's ruin problem comes from

The problem is one of the oldest in probability. Its earliest known mention is in a 1656 letter from Blaise Pascal to Pierre de Fermat, and Christiaan Huygens restated it the following year in De ratiociniis in ludo aleae, one of the first printed works on the maths of chance. Modern casino textbooks still teach it: Hannum and Cabot's Casino Math covers the gambler's ruin in its chapter on gaming maths.

For the wider story of how probability grew out of gambling questions, see our casino probability basics page and the history of casinos.

The fair game: your chance equals start divided by target

Start with a game that has no house edge, such as a coin toss at even money. You bring some money and decide to stop either when you reach a target or when you go broke. What is the chance you reach the target?

There's a neat way to see the answer without heavy algebra. In a fair game, every bet has an average result of zero, so however you play, your average finishing balance equals your starting balance. But you only ever finish in one of two places: at £0 or at your target. If your chance of hitting the target is P, your average finish is P × target. Setting that equal to your start gives P = start ÷ target.

Two things stand out. Modest targets are reached more often, but the losses when you fail are bigger than the gains when you succeed: from £100 aiming for £110, you win £10 about 91% of the time and lose £100 about 9% of the time, which averages out to zero. And this is the best case. Real casino games are not fair, so every figure in this table is an upper limit.

Adding a house edge: a worked roulette example

Now bring in a real bet. You have £20, you want to reach £40, and you bet red or black on a single-zero wheel. Red wins 18 times in 37, so p = 18/37, about 48.65%, and it loses with q = 19/37, about 51.35%.

Option one: one £20 bet. You reach £40 with a single win, so your chance is simply 48.65%.

Option two: £10 bets. You start two wins from the target and two losses from ruin. Two wins in a row take you to £40; two losses in a row take you to £0; a win and a loss in either order put you back at £20 to try again. Only the decisive pairs matter, so your chance is p² ÷ (p² + q²). With p² about 0.2367 and q² about 0.2637, that is about 47.30%.

The Pass line row uses its 1.41% edge: for an even-money bet with no pushes, the win chance is (1 − edge) ÷ 2. In a fair game the number of bets makes no difference. With an edge, splitting the same goal into more bets lowers your chance every time, and the bigger the edge, the bigger the drop. The American roulette row loses more than two and a half points by moving from one bet to £10 bets.

Why more bets make ruin more likely

The roulette example is a small version of a general rule. Your expected loss is the house edge times everything you stake. Small bets aimed at a distant target mean far more total staking before the session ends, so the edge has far more chances to work. Meanwhile, the random swings that might carry you to your goal grow much more slowly: the spread of results rises only with the square root of the number of bets, while the expected loss rises in direct proportion.

Over a short session, luck dominates. Over a long one, the edge does, which is the law of large numbers in action. Seen from the bankroll's side, that is the whole story of ruin: give the edge enough bets, and a finite pot of money runs out. Our page on risk of ruin looks at the same idea from the angle of bankroll size.

This is not advice to bet bigger. Larger bets lose money faster on average, and a single large stake you can't afford is a far worse outcome than a slow session. The point is simply that no pattern of bets escapes the edge.

Why the casino is never the one ruined

The second form of gambler's ruin explains why casinos are comfortable taking bets that can go against them for hours. A player's bankroll is small and finite. The casino's is, by comparison, effectively unlimited, and it takes bets from thousands of players at once. A bad night for the house is absorbed; a bad night for a player can end the session.

Casinos also set a maximum stake on each table, the table limit. That caps how far any single player can push one bet, which matters a great deal for doubling systems. For more on how the margins add up, read how do casinos always win?

Gambler's ruin and the Martingale system

The Martingale is the best-known attempt to dodge ruin: double your stake after every loss so that one win recovers everything plus one unit. It wins small amounts most of the time and loses a large amount occasionally. Here is the sequence for a £5 starting bet on red:

As the Wizard of Odds Martingale answer (opens in a new tab) works out, losing eight even-money bets in a row on a single-zero wheel has a probability of (19/37)^8, about 0.004835, or roughly 1 in 207. That sounds rare, but a player making hundreds of Martingale sequences will meet it. After eight losses the next bet would be £1,280 just to win back £5 profit, and it may be above the table limit. In a Wizard of Odds craps simulation (opens in a new tab), Martingale bettors went broke in 19.65% of sessions, and their ratio of money lost to money bet matched flat betting, approaching the house edge.

Wizard of Odds quotes the mathematician Billingsley: 'No betting system can convert a subfair game into a profitable enterprise.' That sentence is gambler's ruin in a nutshell. Our roulette betting systems page tests the other popular systems against the same maths.

What gambler's ruin means for your own play

None of this is a reason for fear; it's a reason for a plan. Its message is that the only reliable protection against losing everything is to stop. A session with a fixed budget and a clear end point can't run to ruin, because it never reaches the long run where the edge takes everything.

1. Decide in advance the amount you are comfortable losing, and treat it as the price of the session.
2. Set a loss limit equal to that amount, and stop when you reach it.
3. If you like, set a win goal too, and stop if you reach that first.
4. Prefer lower-edge bets such as the Pass line, Banker at baccarat or even-money bets with La Partage.
5. Never raise stakes to win back losses; that is exactly how a limited bankroll meets ruin faster.
6. Use deposit limits and time-outs so the plan holds even when you're tempted to keep going.

A loss limit and a win goal don't change the house edge, but they cap how long the edge gets to work. Our guides to bankroll management and knowing when to stop cover this in more detail, and chasing losses explains why raising stakes after losing is so risky. Free, confidential support is available from the National Gambling Helpline on 0808 8020 133, run by GamCare, 24 hours a day.

Gambler's ruin myths

For why the 'due' feeling is so convincing, see the gambler's fallacy, and for the bigger question, can you make money from casino games? Back on the casino maths hub you'll find the rest of our probability guides.

Gambler's ruin and casino bonuses

A bonus adds money to your balance but doesn't change the edge on any bet, so the maths on this page still applies while you wager. These are mainly slot offers, and table and live games are often excluded or count for less towards wagering, so check the terms. Casumo: new customers aged 18+ who opt in can get a 50% bonus up to £100 on a first deposit of at least £20, wagered 10x within 7 days, with a £5 maximum bet while the bonus is active; game, play and payment restrictions apply.

10bet: new UK customers aged 18+ who choose the casino welcome bonus at sign-up and deposit within 7 days get a 100% match up to £25, to be wagered 10x on slots within 3 days, with bonus winnings capped at £1,000 and Skrill and Neteller deposits excluded. Coral: new players aged 18+ who deposit and bet £10 or more on slots between 1 October 2026 and 1 February 2027 get 100 free spins at 10p each on selected games, to accept within 48 hours and use within 7 days, with no wagering on winnings; Neteller, PayPal, Paysafe, Skrill, prepaid cards and certain debit cards are excluded.

18+. Every casino game has a built-in house edge, so over time you should expect to lose what you stake. Set a budget before you play and stop when it’s gone: safer gambling tools and support. Free, confidential help: National Gambling Helpline 0808 8020 133.