Risk of ruin in gambling: the maths of losing a bankroll
Risk of ruin is the probability that you lose your whole bankroll before you reach a goal or decide to stop. In any casino game with a house edge, it is the number that explains why "playing until I double it" usually ends at zero. This guide gives the formula, tables we computed for dice and roulette, what bet size and volatility change, and what they never change.
What risk of ruin means
Picture a bankroll of 100 units, flat bets of 1 unit, and a target of 200. Each bet moves you one step up or down. Sooner or later you hit 200 or you hit 0. Risk of ruin is the chance of hitting 0 first.
In a perfectly fair game, like a coin flip paying 1 to 1, the answer is simple. Your chance of reaching the target is your bankroll divided by the target, so starting at 100 with a target of 200 you have a 50% chance either way.
Add a small edge against you and the picture changes sharply. The standard result, taught as the gambler's ruin problem in MIT's Mathematics for Computer Science course, is:
P(reach target) = (r^n - 1) / (r^T - 1), where r = q/p
Here p is the chance of winning one even-money bet, q = 1 - p, n is your bankroll in betting units and T is the target. Risk of ruin is 1 minus that figure.
Risk of ruin when trying to double a bankroll
We ran the formula for four even-money games, with flat 1-unit bets and a target of twice the starting bankroll.
| Starting bankroll (units) | Fair coin | Dice 2x at 99% RTP | European roulette, red | American roulette, red |
|---|---|---|---|---|
| 10 | 50.0% | 55.0% | 63.2% | 74.2% |
| 50 | 50.0% | 73.1% | 93.7% | 99.5% |
| 100 | 50.0% | 88.1% | 99.6% | above 99.99% |
| 500 | 50.0% | above 99.99% | above 99.99% | above 99.99% |
The win chances used are 49.5% for the dice bet, 18/37 for red on a single zero wheel and 18/38 on a double zero wheel.
The pattern is the point. A larger bankroll does not protect you here, it makes ruin more certain, because a bigger bankroll means more bets before you get anywhere, and every bet pays the edge. On American roulette, starting with $100 in $1 bets, the chance of reaching $200 is about 1 in 37,650. MIT's notes give the same bound: below 1 in 37,648.
With no target, ruin is certain
The table assumes you stop when you double. Many players do not set a target at all. In that case the maths is blunt: in a game with a negative edge, flat betting continues until the bankroll is gone, with probability 1.
What you can work out is how long it takes on average. For flat 1-unit bets, the expected number of bets before ruin is the bankroll divided by (q - p):
| Game | q - p | Expected bets to lose 100 units |
|---|---|---|
| Dice 2x at 99% RTP | 0.010 | 10,000 |
| European roulette, red | 0.027 | 3,700 |
| American roulette, red | 0.053 | 1,900 |
At a fast crypto dice game, 10,000 bets can pass in an evening with auto-bet. The edge is small per bet but the number of bets is not.
Risk of ruin within a set number of bets
A more practical question is: if I play a fixed number of bets, how likely am I to lose the lot along the way? We computed it step by step for flat 1-unit even-money bets, with no target, stopping only at zero.
| Game and bankroll | Ruined within 1,000 bets | Within 5,000 bets | Within 10,000 bets |
|---|---|---|---|
| Dice 2x at 99% RTP, 50 units | 18.2% | 71.4% | 87.3% |
| Dice 2x at 99% RTP, 100 units | 0.4% | 36.5% | 66.8% |
| European roulette red, 50 units | 34.5% | 95.2% | 99.6% |
| European roulette red, 100 units | 1.7% | 78.9% | 97.9% |
Here a bigger bankroll does help, because there is no target to chase: it buys time. Doubling the bankroll from 50 to 100 units cuts the chance of busting in the first 1,000 bets from 18.2% to 0.4% on dice. But it only delays the outcome. Keep going and every row climbs towards 100%.
To estimate your own figure, convert your bankroll into betting units (bankroll divided by your usual stake), estimate how many bets a session runs to, and read the nearest row. If the bet is not even money, the risk is higher than the table shows, as the next sections explain.
Bet size: what it changes and what it does not
Take $1,000 on European roulette, betting red, with a target of $2,000. We computed the chance of reaching the target, the expected total amount wagered and the expected loss for different flat bet sizes.
| Flat bet | Chance of reaching $2,000 | Expected total wagered | Expected loss |
|---|---|---|---|
| $1 | about 3 in 10^24 | $37,000 | $1,000 |
| $10 | 0.45% | $36,669 | $991 |
| $100 | 36.8% | $9,766 | $264 |
| $500 | 47.3% | $1,999 | $54 |
| $1,000 | 48.6% | $1,000 | $27 |
Two things are true at once. Bigger bets mean fewer bets, so the edge has less time to work and the chance of reaching the target is higher. But the expected loss is still 2.70% of everything you wager, in every row. No bet size gives better than a 48.6% chance of doubling on this wheel, which is just the 18/37 chance of one spin. The house still wins on average in every line of the table.
Small bets do not make a negative game safer either. They make the result more predictable: with $1 bets, losing the whole $1,000 is all but certain.
Volatility and high multipliers
Not every bet pays even money. We solved the same problem for three bets on a 99% RTP game: 100 units, flat 1-unit bets, stop at 200 or 0.
| Bet | Win chance | Chance of reaching 200 | Risk of ruin |
|---|---|---|---|
| 2x dice | 49.5% | 11.9% | 88.1% |
| 10x target | 9.9% | 43.7% | 56.3% |
| 100x target | 0.99% | 41.3% | 58.7% |
Higher volatility reaches the target more often here for the same reason bigger bets do: fewer, larger swings give the edge less time. It does not change the expected loss per unit wagered, and long losing streaks are normal. A 100x bet at 0.99% has about a 37% chance of going 100 bets without a single win.
Without a target the same volatility works against you. With 100 units and no stopping point, the chance of being wiped out within 1,000 flat bets is 0.4% at 2x, 31.9% at 10x and 76.1% at 100x. Bigger swings reach zero sooner.
Systems that raise the stake after a loss, like the Martingale, try to swap a high chance of small wins for a small chance of a very large loss. Our guides on Stake dice strategy and how to beat roulette work through that trade with real table limits. The risk of ruin does not disappear, it is concentrated into one bad run.
Percentage betting and "never going broke"
Betting a fixed percentage of your current bankroll, say 2% per bet, means you never bet your last unit, so you technically never reach zero. In a negative-edge game the bankroll still shrinks towards nothing over time, just in smaller and smaller steps. In practice minimum bet sizes stop you long before that. Our sports betting bankroll management guide covers percentage staking where bettors believe they have an edge, which is a different question from house games.
Provably fair does not change the risk
On a provably fair game you can check that every roll came from seeds fixed before you bet, which our guide on how to verify a provably fair bet explains. That protects you from a casino changing results. It does not change the win chance or the payout, so every number in this guide still applies. A fair game with a 1% edge has exactly the risk of ruin shown in the tables. See what does RTP mean for how the payout sets that edge.
What you can actually control
- Set your loss limit before you start. Treat the bankroll as money you are prepared to spend. If losing it would hurt, it is too large.
- Decide when you stop. Without a stopping point, ruin in a negative-edge game is certain. With one, it is still likely.
- Watch the number of bets, not just the size. Expected loss = total wagered x house edge. Auto-bet and fast games raise the total quickly.
- Prefer a lower edge. At the same bankroll and bet size, 99% dice gives you far more time than double zero roulette.
- Use the tools. A deposit limit or self-exclusion caps the real downside. Our responsible gambling page lists where to get help.
Sources
- MIT 6.042J Mathematics for Computer Science: Gambler's Ruin (A. R. Meyer, 2013)
- Mathematics for Computer Science (Lehman, Leighton and Meyer): Gambler's Ruin, LibreTexts edition
- theScore Bet: Euro Roulette rules (Evolution)
- Gamdom: Gamdom Originals (RTP around 99%)
- All tables computed by us in Python: closed-form gambler's ruin and expected duration for even-money bets, and an absorbing Markov chain for the 10x and 100x bets
Key terms
- House edge
- The average percentage of every wager that a game is built to keep for the casino.
- Return to player (RTP)
- The share of all money staked that a game is designed to return to players across a very large number of rounds.
- Volatility
- How sharply a game's results swing: high volatility brings rare big wins, low volatility brings frequent small ones.
- Dice
- Seed-verifiable game where you set a target number and predict if the random roll lands above or below it.
- Win chance
- The probability that a bet wins, shown by many provably fair games next to the payout multiplier.
- Target multiplier
- The multiplier a player chooses in limbo or auto cash-out crash; the bet wins if the result reaches it.
- Provably fair
- A cryptographic method that lets you confirm a game result was set before your bet and left untouched afterwards.
- Deposit limit
- A ceiling you choose on the amount you can deposit to one casino account per day, week or month.
- Self-exclusion
- A voluntary block you put on your own gambling, at one operator or across a whole regulated market, for a fixed period.