What is a risk of ruin calculator?
A risk of ruin calculator estimates the chance that a losing run uses up your whole bankroll before you stop. With a bankroll of 1,000 units, a stake of 20 units, even-money odds (2.00), a 52% win chance and 500 bets, the chance of hitting zero within those bets is about 0.25%. If you never stopped, it would be about 1.8%. Those numbers describe a small edge and a small stake. Change either and the answer moves a lot.
The calculator is useful for two opposite readers. If you believe you have an edge, it shows how big a bankroll protects you from an unlucky start. If you do not have an edge, which is the case for every casino game, it shows what the maths already says: the longer you play, the more certain the loss, and a larger bankroll only delays the end. That second reading is the one the page spends the most time on.
How do you use the risk of ruin calculator?
- Enter your bankroll. This is the total you could lose before you must stop, in units.
- Enter your stake per bet. The calculator assumes the same stake every time.
- Enter the odds you get. Decimal, fractional and American formats all work.
- Enter your estimated win chance. For a casino game use the true chance from the rules, such as 47.37% for an even-money roulette bet.
- Enter the number of bets. It starts at 500.
- Read the large number. It is the chance of reaching zero within your number of bets. The panel gives the chance if you never stop and, where it applies, what bankroll would cut the risk.
Example: Bankroll 1,000, stake 20, odds 2.00, chance 52%, 500 bets. The edge per bet is 0.04 x 20 = +0.80 units. The chance of ruin within 500 bets is 0.25%. Never stopping, it is 1.8%.
How is the risk of ruin calculated?
Each bet is treated as a small random step with a mean (the expected value, mu) and a variance (s squared). Over many bets the total behaves like a random walk with drift. For a bankroll B, the chance of ever reaching zero when mu is positive is exp(-2 x mu x B / s squared). If mu is zero or negative, ruin is certain given enough bets. For a limited number of bets N, the calculator uses the standard first-passage result for a random walk with drift: Phi((-B – mu N) / (s root N)) + exp(-2 mu B / s squared) x Phi((-B + mu N) / (s root N)), where Phi is the normal distribution function.
This is an approximation. It treats the bankroll as continuous, so it ignores the exact size of the last bet. For even-money bets we checked it against an exact step-by-step calculation, and it agrees to within about 0.01 percentage points. For long odds, with rare large wins, it is rougher. The gambler’s ruin problem is the classic exact version for fixed steps.
| Step | Working | Result |
|---|---|---|
| Mean per bet (mu) | 0.52 x 20 – 0.48 x 20 | +0.80 |
| Variance per bet | 400 – 0.64 | 399.36 |
| Ruin if never stopping | exp(-2 x 0.80 x 1000 / 399.36) | 1.8% |
| Ruin within 500 bets | first-passage formula | 0.25% |
Why does a bigger bankroll not save you from a house edge?
Every casino game has a negative expected value per bet. With a negative mu, the chance of ruin if you keep playing is 100%, whatever the bankroll. A bigger bankroll only pushes ruin further out. The table uses an even-money roulette bet in a double-zero game (47.37% chance, a house edge of 5.26%) with a stake of 20 units. Read across a row to see the risk grow with the number of bets, and down a column to see the bankroll help less and less.
| Bankroll | Ruin within 500 bets | Ruin within 5,000 bets | Ruin within 50,000 bets |
|---|---|---|---|
| 200 | 91.7% | 100.0% | 100.0% |
| 500 | 67.4% | 100.0% | 100.0% |
| 1,000 | 20.6% | 100.0% | 100.0% |
| 2,000 | 0.08% | 99.5% | 100.0% |
| 5,000 | 0.00% | 62.7% | 100.0% |
A 5,000-unit bankroll looks safe over 500 bets, but over 50,000 bets it is close to certain ruin. Real sessions are rarely that long, yet the table shows why “I have enough money to keep going” is not a plan. The expected loss grows with every bet, and the roulette odds and payouts page shows where the 5.26% comes from.
What does a small edge do to the risk?
If you do have an edge, the bankroll decides how likely you are to survive the early losses. The table keeps the stake at 20 units, the odds at 2.00 and the bankroll at 1,000, and changes only the win chance. At 50% the edge is zero and ruin is certain if you never stop. At 52% there is a 1.8% chance of ruin ever, and at 55% the risk is negligible.
| Win chance | Ruin within 500 bets | Ruin if never stopping |
|---|---|---|
| 50% | 2.5% | 100.0% |
| 51% | 0.86% | 13.5% |
| 52% | 0.25% | 1.8% |
| 55% | 0.00% | 0.00% |
For the default 52% edge, the calculator also shows the bankroll that keeps the lifetime risk at 5%, 2% or 1%: about 748, 976 and 1,149 units. These are for the edge you typed in, so they are only as good as your estimate. A real edge of 51% instead of 52% roughly doubles the bankroll you need.
How much does the stake matter?
Stake size matters more than most people expect. With the same 1,000 bankroll and a 52% edge, raising the stake from 20 to 50 units multiplies the lifetime ruin risk many times over, because the bankroll covers fewer bets.
| Stake | Bets the bankroll covers | Ruin within 500 bets | Ruin if never stopping |
|---|---|---|---|
| 10 | 100 | 0.00% | 0.03% |
| 20 | 50 | 0.25% | 1.8% |
| 50 | 20 | 13.7% | 20.1% |
| 100 | 10 | 39.2% | 44.9% |
The edge per bet falls in step with the stake, but the swings fall faster. That is why stake sizing methods such as the Kelly criterion exist, and why they come with an honest warning about estimate errors.
What does the calculator leave out?
- Unequal stakes. It assumes the same stake every bet. If you bet more after losses, ruin comes sooner.
- Estimate errors. A win chance that is too high gives a risk that is too low.
- Bet limits and fees. Table limits, commission and bonuses are not modelled.
- Correlated bets. Bets on the same event are not independent.
- The last bet. It ignores that you cannot place a bet larger than your remaining bankroll.
- Your limits. It ignores the point at which you would choose to stop.
What mistakes do people make with risk of ruin?
- Treating a low short-term number as safe. 500 bets can look fine when 50,000 do not.
- Using the bankroll as a target to lose. The point of the number is to see how likely loss is, not to plan for it.
- Using an advantage you do not have. Casino games do not give players a positive edge, so the answer for them is always that loss is certain over enough bets.
- Chasing losses. Raising the stake after a loss shortens the number of bets the bankroll covers.
What should you try next?
Change the number of bets from 500 to 5,000 and watch the within-N risk approach the lifetime figure. Then halve the stake and see the effect. For the average result of the same bet, use the expected value calculator. If you are thinking about how much to stake, the Kelly criterion calculator shows why fractional stakes are used. To set limits before you play, see the Safer play toolkit.
How to read the result
The large number is the estimated chance of reaching zero within your number of bets. The panel shows the chance of ruin if you never stopped. When the edge is positive it adds the bankroll that would hold the lifetime risk to 5%, 2% and 1%. When the edge is zero or negative it shows how many bets the bankroll lasts on average, because ruin is certain given enough bets. A low short-run number is not safety if the edge is negative.
How it is calculated
Each bet has a mean result mu = chance x stake x (d - 1) - (1 - chance) x stake and a variance s squared. Chance of ruin if you never stop = exp(-2 x mu x bankroll / s squared) when mu is positive, otherwise 100%. Within N bets = Phi((-B - mu N) / (s root N)) + exp(-2 mu B / s squared) x Phi((-B + mu N) / (s root N)). This is a random-walk approximation that treats the bankroll as continuous.
Worked example
Worked example
Bankroll 1,000, stake 20, odds 2.00, chance 52%, 500 bets. mu = +0.80 and s squared = 399.36. Chance of ruin never stopping = exp(-2 x 0.80 x 1000 / 399.36) = 1.8%. Chance of ruin within 500 bets = 0.25%. Bankroll for a 5% lifetime risk: about 748.
Limits
This is a normal-curve approximation, close for even-money bets and rougher for long odds. It assumes a constant stake, independent bets and a win chance that is exactly right. It ignores table limits, fees, bonuses, tax and the choice to stop. A small error in the win chance changes the result a lot. A house-edge game always shows certain ruin over enough bets. Results are estimates, not predictions.