The Risk of Ruin Formula
Risk of ruin is the probability that you will lose your entire bankroll before recovering from a downswing. For any poker player with a positive win rate, the risk of ruin over an infinite time horizon is a single number between 0% and 100%. The lower it is, the safer your bankroll.
The formula that governs this probability comes from the mathematics of random walks with positive drift. It applies to any repeated game where you have a small edge and face variance on every hand.
- WR
- Win rate in bb/100 hands
- BR
- Bankroll in big blinds
- SD
- Standard deviation in bb/100 hands
- e
- Euler’s number (~2.718)
This formula calculates the probability of going broke over an infinite time horizon, assuming you play at fixed stakes and maintain a positive win rate. If your win rate is zero or negative, the risk of ruin is 100% — you will go broke eventually regardless of bankroll size.
Worked Examples
Three scenarios that cover the range of situations most cash game players face. Each example plugs real numbers into the formula so you can see how win rate, bankroll, and standard deviation interact.
RoR = e(-30,000 / 7,225)
RoR = e(-4.15)
RoR = e(-9,000 / 8,100)
RoR = e(-1.11)
RoR = e(-40,000 / 19,600)
RoR = e(-2.04)
Bankroll Requirements Table
You can invert the risk of ruin formula to answer a more practical question: how large does your bankroll need to be to keep your risk of ruin below a specific threshold?
Solve for the bankroll (in big blinds) needed to stay below your chosen risk of ruin. Divide the result by your buy-in size to get the number of buy-ins required.
| Win Rate | SD | 5% RoR | 2% RoR | 1% RoR |
|---|---|---|---|---|
| 1.0 bb/100 | 80 | 96 buy-ins | 125 buy-ins | 148 buy-ins |
| 2.0 bb/100 | 85 | 54 buy-ins | 71 buy-ins | 84 buy-ins |
| 3.0 bb/100 | 85 | 36 buy-ins | 48 buy-ins | 56 buy-ins |
| 5.0 bb/100 | 90 | 25 buy-ins | 32 buy-ins | 38 buy-ins |
| 3.0 bb/100 | 140 (PLO) | 98 buy-ins | 128 buy-ins | 151 buy-ins |
PLO players need roughly 3x the bankroll of NLHE players at the same win rate because of the higher standard deviation. The SD in PLO typically runs between 120 and 160 bb/100, compared to 75 to 95 bb/100 in NLHE 6-max. That difference in variance dominates the bankroll calculation because SD is squared in the formula.
Risk of Ruin Calculator
Enter your numbers below. The calculator applies the exact formula from the worked examples above and shows the minimum bankroll needed to hit common risk thresholds.
What the Formula Does Not Tell You
The risk of ruin formula is a useful baseline, but it makes several assumptions that do not hold in practice. Understanding where the formula breaks down is as important as knowing how to use it.
First, the formula assumes an infinite time horizon. It calculates the probability that you will ever go broke, given unlimited play. In reality, you play a finite number of hands. A player with a 5% risk of ruin over infinite hands has a lower risk over any bounded session. For specific session lengths, Monte Carlo simulation provides more accurate estimates than the closed-form formula.
Second, the formula assumes your win rate stays constant. It does not. Games change, player pools shift, and your own skill improves or deteriorates over time. A win rate measured over 100,000 hands at NL50 six months ago may not reflect your current edge. The formula treats win rate as a known constant, but in practice it is an estimate with its own uncertainty.
Third, the formula assumes fixed stakes. Most players move down in stakes when their bankroll shrinks and move up when it grows. This adaptive behavior reduces actual risk of ruin significantly compared to what the formula predicts. A player who drops from NL100 to NL50 after losing 15 buy-ins has effectively doubled their remaining bankroll in buy-in terms.
Finally, the formula ignores deposits and withdrawals. If you add money to your bankroll during a downswing, your risk of ruin is lower than the formula suggests. If you withdraw profits regularly, it is higher. For a more complete picture that accounts for these factors, run simulations with specific hand counts and withdrawal schedules using PrimeDope’s variance calculator.
Risk of Ruin for Tournament Players
The formula above applies to cash games. Tournament poker has a fundamentally different variance profile because of the top-heavy payout structure. In a cash game, you win or lose a relatively small number of big blinds each hand. In a tournament, you risk your entire buy-in every time you register, and the vast majority of your lifetime profit comes from a small number of deep runs and final table finishes.
Tournament standard deviations are typically 3 to 5 times higher than cash game SDs when measured in buy-in units. A cash game player might have an SD of 85 bb/100 (roughly 0.85 buy-ins per 100 hands). A tournament player with a similar edge might have an SD of 3 to 5 buy-ins per tournament. This is why tournament bankroll recommendations start at 100 buy-ins and often extend past 200 for high-variance formats like large-field MTTs.
You can still apply the risk of ruin concept to tournaments, but the inputs change. Your win rate becomes your ROI per tournament (expressed as a fraction of the buy-in), and your SD is measured per tournament rather than per 100 hands. For tournament-specific calculations, use PrimeDope’s tournament variance calculator, which models the actual payout structures and field sizes you face.