Risk of Ruin, Explained
After reading this you can compute the chance that a fixed bankroll busts before it reaches a target, understand why a small edge against you turns almost certain ruin, and avoid the classic mistakes that make the number look better than it is.
What risk of ruin measures
Risk of ruin is the probability that you lose your entire bankroll while betting. You start with some number of units, you bet one unit at a time on an even-money proposition, and you keep going until you either hit zero (ruin) or reach a target you set in advance.
Here is the hook. Suppose you play a game where you win each bet 49% of the time, and you bring 50 units. That is a tiny disadvantage, close to a fair coin. If you decide to play forever, your chance of eventually going broke is about 0.8636, or roughly 86%. A 1-point gap in win probability, repeated thousands of times, almost guarantees ruin. This calculator makes that concrete.
The classic version assumes flat even-money bets: every wager is one unit, and a win returns one unit. Varying your stake or the payout changes the math entirely. This model is the baseline everything else is measured against.
When to use it, and when not
Use this model when your bets are close to even money, roughly the same size, and independent of each other. A pass-line bet in craps, a red-black bet in roulette, or a coin-flip game all fit reasonably well. The model answers one question cleanly: given this edge and this bankroll, what fraction of players bust?
Do not use it when the payouts are uneven, the stakes change, or the bets are correlated. A Martingale sequence, where you double after each loss, breaks every assumption here. So do slot machines and lotteries, which pay large multiples rarely. For those, model the actual bet structure directly.
The single-unit assumption matters more than people expect. Betting 5 units at a time on a 50-unit roll behaves like a 10-unit roll, not a 50-unit one. Fewer, larger bets means higher variance and a higher chance of ruin.
The formula and the intuition
Let p be your probability of winning one bet, and q = 1 - p the probability of losing. Write n for your starting bankroll in units and N for the target you want to reach before busting.
When the game is not fair (p \ne 0.5), the probability of reaching the target N before hitting zero is:
Here (q/p) is the ratio of losing odds to winning odds. If you are at a disadvantage, q/p \gt 1 and that ratio raised to a power grows fast. The risk of ruin is just one minus the chance of reaching the target:
When you play forever there is no target, so let N \to \infty. If you have an edge (p \gt 0.5, so q/p \lt 1), the formula collapses to a clean result:
Every extra unit of bankroll multiplies your ruin chance by q/p. With a real edge, ruin drops geometrically. But if you are at a disadvantage and play forever, q/p \gt 1, the ratio raised to a large power blows up past 1, and the formula caps at R = 1. Ruin is certain. That is why the "play forever" case with any house edge always returns 1 in the limit, and why finite targets are the only way to see numbers below 1 when you are behind.
Worked example: 49% win, 50 units, no target
Reproducing the demo
Load the demo values: prob = 49, units = 50, target = 0. Target zero means play until ruin or until you decide to stop, with no upper bound.
- Convert the win probability: p = 0.49, so q = 0.51.
- Form the ratio: q/p = 0.51 / 0.49 = 1.040816.
- You are at a disadvantage, so use the finite-target formula with a very large target. Playing truly forever gives R = 1 in the limit, so this calculator reports the eventual-ruin figure for the practical horizon.
- For a concrete comparison, set a target of 100 units. Then R = 1 - \frac{1 - 1.040816^{50}}{1 - 1.040816^{100}}.
- Compute the powers: 1.040816^{50} \approx 7.398 and 1.040816^{100} \approx 54.73.
- Plug in: R = 1 - \frac{1 - 7.398}{1 - 54.73} = 1 - \frac{-6.398}{-53.73} = 1 - 0.1191 = 0.8809.
So doubling 50 units to 100 against a 49% game busts about 88% of the time. Playing with no ceiling at all pushes that toward certainty. The lesson is blunt: a house edge that small still beats a bankroll that large.
How the edge flips the picture
The disadvantage case looks grim, so it helps to see what happens when the edge is yours. Suppose p = 0.52 and you play forever. Then q/p = 0.48/0.52 = 0.9231, and ruin is R = 0.9231^n.
| Bankroll (units) | q/p | Risk of ruin |
|---|---|---|
| 10 | 0.9231 | 0.4494 |
| 25 | 0.9231 | 0.1350 |
| 50 | 0.9231 | 0.01822 |
| 100 | 0.9231 | 0.0003319 |
| 200 | 0.9231 | 0.0000001102 |
With an edge, bankroll is your friend. At 50 units the ruin chance is under 2%, and at 100 units it is about 3 in 10,000. The same geometric growth that dooms the disadvantaged player rescues the advantaged one. This is why edge and bankroll together, not either alone, decide survival.
Reading and interpreting the result
The output is a probability between 0 and 1. Read it as the fraction of identical players who would bust under these exact rules. A ruin of 0.30 means 3 in 10 players go broke and 7 in 10 reach the target.
Two numbers control everything: the sign of your edge and the size of your bankroll relative to the target. If the edge is negative, adding bankroll only delays ruin, it does not prevent it. If the edge is positive, bankroll drives ruin down fast, and there is a point past which further bankroll barely helps because ruin is already tiny.
- Bankroll (n)
- Units you start with. Each unit is one flat bet.
- Target (N)
- Units you want to reach before busting. Zero means no ceiling.
- Edge
- The gap between your win probability and 0.5. Positive is an advantage, negative a disadvantage.
Common mistakes
The first mistake is treating uneven payouts as even money. Roulette pays 1 to 1 on red, but the green pockets make p \approx 0.4737 on a single-zero wheel, not 0.5. Enter the real win probability, not the payout ratio.
The second mistake is mixing up units and money. If one unit is 20 dollars and you have 1,000 dollars, your bankroll is 50 units, not 1,000. Always convert to bet-sized units first.
The third mistake is expecting the "play forever" case to give a comfortable number when you are behind. It cannot. Against any house edge, unbounded play always ends in ruin. Set a realistic target if you want a probability below 1.
The fourth mistake is assuming a bigger stake is safer because it "gets there faster." Larger bets shrink your effective unit count and raise variance, so ruin goes up, not down.
Related tools
Once you know your ruin, decide how big to bet. The Kelly Criterion Calculator gives the stake that maximizes long-run growth when you have an edge. If you are tempted to chase losses by doubling, the Martingale Simulator shows the true bust probability of that system, which this even-money model deliberately does not cover. To see how long losing runs actually get, use the Streak Probability tool. And if your bankroll comes from a casino promotion, the Bonus Wagering EV calculator tells you what it is worth after the rollover requirement.
Frequently asked questions
Why is my risk of ruin 100% when I set no target?
Because your win probability is 0.5 or below. With any disadvantage and unlimited play, ruin is mathematically certain over an infinite horizon. Set a finite target to see a number below 1.
Does a bigger bankroll always reduce ruin?
Only when you have an edge. At p = 0.52, going from 25 to 50 units drops ruin from 0.135 to 0.018. At p = 0.49 with no target, extra units only slow the inevitable.
How do I convert my money into units?
Divide your bankroll by your flat bet size. A 500 dollar roll at 25 dollars a bet is 20 units. Enter 20, not 500.
Can I use this for blackjack card counting?
Roughly, if you flat-bet. Counting means varying your bet with the count, which changes both edge and stake per hand, so the flat-bet formula underestimates variance. Treat the result as an optimistic bound.
What win probability does even-money roulette have?
On a single-zero wheel, a red or black bet wins 18 of 37 spins, so p \approx 0.4865. On a double-zero wheel it is 18 of 38, or p \approx 0.4737. Both guarantee ruin over unlimited play.
Related tools
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Bonus Wagering EV
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