Risk of ruin in betting is the chance that losses take your whole bankroll before your edge has time to pay off. It rises with your stake size and the length of your prices, and falls as your edge grows. With illustrative numbers, $2.00 bets with a true 52% chance (a 4% edge) and a bankroll of 25 stakes carry about a 13.5% risk of ruin if you keep betting; with 50 stakes it is about 1.8%.
Every formula below assumes your edge is real and exactly known. It never is: your true edge, not the planned one, sets the risk you face, and with no edge ruin is certain if you keep betting.
What risk of ruin measures
Risk of ruin answers one question about a staking plan: if you keep following it, what is the chance the bankroll can no longer cover the next bet? Three inputs decide the answer:
- Your betting edge per $1 staked: true chance x decimal odds - 1.
- How far each bet swings, which grows with the price, as variance in betting shows.
- How many stakes the bankroll holds: bankroll / stake. A $1,000 bankroll betting $20 a time holds 50 stakes.
Ruin "ever, if you keep betting" is likelier than ruin within one season, and ruin can mean $0 or a floor you set in advance, such as a quarter of the starting bankroll.
The risk of ruin formula
The exact formula for $2.00 bets
For flat stakes at $2.00, the classic gambler's ruin result gives the answer exactly, betting with no end:
risk of ruin = ((1 - edge) / (1 + edge)) to the power of the stakes held
With a 52% chance, edge = 0.52 x 2.00 - 1 = 0.04. A bankroll of 25 stakes gives (0.96 / 1.04) to the power 25 = 0.9231 to the power 25 = 0.135, or 13.5%. A bankroll of 50 stakes gives 0.9231 to the power 50 = 0.018, or 1.8%.
An approximate formula for any price
risk of ruin = e to the power of (-2 x edge x stakes held / variance), approximately
Here variance = odds x odds x p x (1 - p) for a $1 bet, p is your true chance of winning, and e is the constant 2.718, the e^x key on a phone calculator. On the same $2.00 bets: variance = 2 x 2 x 0.52 x 0.48 = 0.9984, the exponent is -2 x 0.04 x 25 / 0.9984 = -2.003, and e to the power -2.003 = 0.135. The approximation matches the exact 13.5%.
With a small edge, variance is close to odds - 1, which can stand in for it. Because edge and stakes held sit side by side in the exponent, halving either one takes the square root of the risk.
What the formulas assume
Both are guides to scale. They assume:
- You bet forever. Over a fixed number of bets the risk is lower.
- Every stake is the same size. Percentage staking has its own formula, below.
- The edge is exact and the same on every bet.
- Bets are independent. Two bets on the same result behave like one bigger bet.
Long prices and small bankrolls make the approximation rougher, though in the flat-stake examples here it lands within about 2 percentage points of an exact count.
How stake size changes risk of ruin
Example: Illustrative numbers, not a real market. A $1,000 bankroll and bets at $1.90, such as line or total markets, where your true chance is 54.2% (1.03 / 1.90 = 0.542), a 3% edge. Variance per $1 = 1.90 x 1.90 x 0.542 x 0.458 = 0.896.
| Stake | Stakes held | Expected profit per bet | Ruin within 1,000 bets | Ruin if you keep betting (formula) |
|---|---|---|---|---|
| $10 | 100 | $0.30 | Under 0.01% | 0.12% |
| $20 | 50 | $0.60 | 1.4% | 3.5% |
| $40 | 25 | $1.20 | 14.6% | 18.7% |
| $100 | 10 | $3.00 | 49.4% | 51.2% |
The working for the $40 row: 2 x 0.03 x 25 / 0.896 = 1.674, and e to the power -1.674 = 0.187. The 1,000-bet column is an exact count over every way 1,000 bets can land, with ruin taken as the bankroll no longer covering one stake.
Each doubling of the stake doubles the expected profit per bet and takes the square root of the risk: 3.5% at $20 becomes 18.7% at $40, because the square root of 0.0351 is 0.187. At $100 a bet, survival is close to a coin toss.
Risk: Betting involves risk. A smaller stake lowers the chance of ruin but never removes it, and every row here assumes a 3% edge that ordinary bookmaker prices do not give. See responsible gambling for limits and support.
Longer prices need more stakes in the bankroll
At the same edge a longer price swings harder, so the same bankroll is less safe. With illustrative prices, each with a 3% edge (true chance = 1.03 / odds):
| Price | True chance | Variance per $1 | Ruin with 50 stakes (formula) | Stakes for about a 5% risk |
|---|---|---|---|---|
| $2.00 | 51.5% | 1.00 | 5.0% | 50 |
| $3.00 | 34.3% | 2.03 | 22.8% | 101 |
| $6.00 | 17.2% | 5.12 | 55.7% | 256 |
| $11.00 | 9.4% | 10.27 | 74.7% | 513 |
The last column uses e to the power -3, about 5%, so stakes needed = 1.5 x variance / edge. For $6.00: variance = 6 x 6 x 0.1717 x 0.8283 = 5.12, and 1.5 x 5.12 / 0.03 = 256 stakes. An exact count gives 22.5%, 55.1% and 74.3% for the 50-stake column at $3.00, $6.00 and $11.00, close to the formula.
Backing $11.00 chances takes about ten times the stakes in the bankroll that $2.00 bets need for the same risk. The long straight runs behind that are set out in losing streaks in betting.
Why an overestimated edge raises the real risk
The formulas take your edge as given, so a plan built on too high an edge understates the real risk. With illustrative numbers, $2.00 bets and a bankroll of 25 stakes, planned on a 4% edge:
| True edge | True chance | Ruin within 1,000 bets | Ruin if you keep betting |
|---|---|---|---|
| +4%, as planned | 52% | 11.2% | 13.5% |
| +2% | 51% | 23.8% | 36.8% |
| 0% | 50% | 42.9% | Certain, given enough bets |
| -4% | 48% | 83.0% | Certain, after about 625 bets on average |
Halving the edge halves the exponent, which takes the square root of the risk: the square root of 0.135 is about 0.37, and the exact figure for a 2% edge is 36.8%. With a negative edge, flat stakes lose about the edge x the stake on each bet on average, so 25 stakes last about 25 / 0.04 = 625 bets.
An overestimate also raises the stake. The Kelly criterion sizes bets in proportion to the edge, so a 4% estimate on a true 2% edge stakes twice the true Kelly share, where long-run growth falls to about zero. Fractional Kelly covers that trade.
Edges get overestimated by fair prices that keep part of the margin, by too few bets (sample size in betting), by prices that move before a bet is accepted and by backtests scored on the data they were tuned on. Beating the closing line is the earliest check that an edge is real.
Going broke betting at a negative edge
Bets at ordinary bookmaker prices usually carry a negative edge, because the margin is built in. Take $1.90 on a two-way market where each side's true chance is 50%: edge = 0.50 x 1.90 - 1 = -5%. With illustrative numbers, a $1,000 bankroll staked at $20 a bet (50 stakes) is gone:
| By this many bets | Chance the bankroll is gone |
|---|---|
| 500 | 18% |
| 1,000 | 62% |
| 2,000 | 93% |
| 3,000 | 99% |
On average it lasts about 50 / 0.05 = 1,000 bets. Smaller stakes only stretch the time: at $10 a bet it lasts about 2,000 bets and ends the same way. Topping up a lost bankroll starts the count again, and raising stakes to win it back is chasing losses, which gets there sooner.
Bankroll risk of ruin with percentage staking
Staking a fixed share of your current bankroll shrinks every stake after a loss, so the bankroll never quite reaches zero. Ruin then means falling to a floor you choose, and an approximation for the chance of ever reaching it is:
chance of ever falling to a share x of the starting bankroll = x to the power of (2 x edge / (stake share x variance) - 1)
With the illustrative $1.90 bets at a 3% edge (variance 0.896) and a floor of a quarter of the starting bankroll:
| Stake share | Compared with full Kelly (3.3%) | Chance of ever falling to 25% |
|---|---|---|
| 1% | 0.3 x Kelly | 0.04% |
| 2% | 0.6 x Kelly | 3.9% |
| 3.3% | Full Kelly | About 25% |
| 5% | 1.5 x Kelly | 62.5% |
| 7% | 2.1 x Kelly | Certain over time |
The working for 2%: 2 x 0.03 / (0.02 x 0.896) - 1 = 2.348, and 0.25 to the power 2.348 = 0.039. Full Kelly here is edge / (odds - 1) = 0.03 / 0.90 = 3.3%, which the Kelly criterion calculator works out for any price. Above about twice that share the exponent turns negative, so any floor is reached in time.
In practice the plan ends before zero, once a stake falls below what a bookmaker accepts. Staking plans compares flat and percentage staking, and bankroll management covers setting the bankroll.
Risk: Betting involves risk. These chances assume a real 3% edge; at a negative edge, a shrinking stake only slows the losses. See responsible gambling for limits and support.
Mistakes that raise your risk of ruin
With the illustrative numbers above:
| Mistake | What it does to the risk |
|---|---|
| Planning with the edge you hope for | A true 2% instead of a planned 4% turns 13.5% into 36.8% (25 stakes at $2.00) |
| Keeping the stake after the bankroll falls | $40 stakes on a bankroll down from $1,000 to $600 hold 15 stakes, not 25: about 37% instead of 18.7% at $1.90 |
| Counting linked bets as separate | Two $40 bets on the same result act like one $80 stake, so $1,000 holds 12.5 stakes: about 43% |
The working for the second row: e to the power of (-0.06696 x 15) = e to the power -1.004 = 0.366, where 0.06696 = 2 x 0.03 / 0.896.
Set the bankroll before you start, from money you can afford to lose, and choose deposit limits that together stay within it. Under the National Consumer Protection Framework, Australian online bookmakers must offer a deposit limit: a decrease applies at once, and an increase only after 7 days. Staking only protects an edge; finding one is the work of a betting strategy.
Risk: Betting involves risk. Every figure here assumes an edge that is real and known, ordinary bookmaker prices usually give a negative one, and then ruin is only a matter of time. There is no guarantee of profit. See responsible gambling for limits and support.