Bankroll management for a winning poker player is not just a rule-of-thumb problem. If you have a positive win rate, a measurable standard deviation, and a fixed-stakes plan, you can estimate the probability of going broke before long-term expectation has time to work.
This calculator uses a fixed-stakes exponential approximation for risk of ruin:
Here, μ is your win rate, B is your bankroll, and σ² is variance. In poker terms, this means win rate, bankroll size, and the square of your standard deviation. The result is a probability between 0 and 1 — multiply by 100 to read it as a percentage.
Poker Risk of Ruin Calculator
How the Formula Works
The calculator estimates the chance that a bankroll eventually reaches zero under a simplified fixed-stakes model. It treats your poker results as a random process with two opposing forces:
- Positive drift: your long-term win rate pushes the bankroll upward.
- Variance: short-term results create swings around that upward trend.
The approximation can be written as:
The output is exact for the simplified continuous model and approximate for real poker results. For example, an output of 0.1353 corresponds to an estimated 13.53% probability of going broke under fixed-stakes assumptions.
Variable Definitions
| Symbol | Meaning | Poker Input | Typical Unit |
|---|---|---|---|
| μ (mu) | Drift / expected gain | Win rate | bb/100 hands |
| σ (sigma) | Standard deviation | Volatility | bb/100 hands |
| σ² | Variance | Standard deviation squared | (bb/100)² |
| B | Starting capital | Bankroll | big blinds |
| RoR | Risk of ruin | Bust probability | 0 to 1, or percent |
Important: if your win rate is zero or negative, fixed-stakes risk of ruin approaches 100% over an infinite time horizon. This model is only meaningful for players with a positive expected edge.
Spreadsheet and Code Formulas
Use the same units for win rate, standard deviation, and bankroll. If your win rate and standard deviation are expressed per 100 hands, your bankroll should be expressed in big blinds.
=EXP(-2 * A1 * B1 / (C1^2))
=(-C1^2 * LN(D1)) / (2 * A1)
import math
def ror(winrate, bankroll, stddev):
return math.exp(-2 * winrate * bankroll / stddev**2)
Example: with win rate 5 in A1, bankroll 2000 in B1, and standard deviation 100 in C1, the formula =EXP(-2*A1*B1/(C1^2)) returns 0.1353, or an estimated 13.53% probability of going broke.
Example: Why Variance Hurts Winning Players
Consider a solid winning No-Limit Hold’em cash player with the following numbers:
- Win rate: 5 bb/100
- Standard deviation: 100 bb/100
- Bankroll: 2,000 bb, or 20 full 100bb buy-ins
RoR = exp(-20000 / 10000)
RoR = exp(-2)
RoR = 0.1353 = 13.53%
The player is a clear long-term winner, but a 20-buy-in bankroll still produces an estimated 13.53% bust probability under the fixed-stakes model. To reduce that estimate to about 1%, the same player would need approximately 4,605 big blinds, or roughly 46 full buy-ins.
This is the practical takeaway: variance alone can break a profitable player whose bankroll is too thin relative to their standard deviation.
Bankroll Tables (Quick Reference)
These tables use the same fixed-stakes approximation. They are useful for quick checks of where your bankroll stands, but they do not replace a full variance simulation or a realistic bankroll plan that accounts for moving down in stakes.
Table 1: Win Rate = 5 bb/100, Standard Deviation = 100 bb/100
| Bankroll (BB) | Buy-ins (100bb) | Estimated Bust Probability |
|---|---|---|
| 1,000 | 10 | 36.79% |
| 2,000 | 20 | 13.53% |
| 3,000 | 30 | 4.98% |
| 4,000 | 40 | 1.83% |
| 4,605 | 46 | 1.00% |
| 5,000 | 50 | 0.67% |
| 6,000 | 60 | 0.25% |
Table 2: Win Rate = 3 bb/100, Standard Deviation = 100 bb/100
Lower win rate, same variance — more conservative estimates appropriate for micro and small-stakes regulars.
| Bankroll (BB) | Buy-ins (100bb) | Estimated Bust Probability |
|---|---|---|
| 2,000 | 20 | 30.12% |
| 3,000 | 30 | 16.53% |
| 4,000 | 40 | 9.07% |
| 5,000 | 50 | 4.98% |
| 7,000 | 70 | 1.50% |
| 7,675 | 77 | 1.00% |
| 10,000 | 100 | 0.25% |
Table 3: Bankroll Required for 1% Estimated Bust Probability
| Win Rate | σ = 80 bb/100 | σ = 100 bb/100 | σ = 120 bb/100 |
|---|---|---|---|
| 2 bb/100 | 7,368 BB (74 BI) | 11,513 BB (115 BI) | 16,579 BB (166 BI) |
| 3 bb/100 | 4,912 BB (49 BI) | 7,675 BB (77 BI) | 11,052 BB (111 BI) |
| 5 bb/100 | 2,947 BB (30 BI) | 4,605 BB (46 BI) | 6,631 BB (66 BI) |
| 8 bb/100 | 1,842 BB (18 BI) | 2,878 BB (29 BI) | 4,145 BB (41 BI) |
Note: BI = full 100bb buy-ins. Typical NLHE standard deviation falls around 80–120 bb/100 depending on format and style; PLO can be higher.
Mathematical Background
The expression exp(-2μB/σ²) comes from the mathematics of a random process with positive drift and volatility. In poker language, the drift is your win rate, and the volatility is your standard deviation.
If X(t) represents a simplified bankroll process, the model can be written as:
Where μdt represents the upward drift over time, and σdW represents random fluctuations from a Wiener process. This is not a claim that poker results literally follow continuous Brownian motion. It is a useful approximation that turns out to behave well in practice and is easier to work with than a hand-by-hand model.
For a process starting at B with an absorbing barrier at zero (going broke), the probability of ever hitting zero before reaching infinity is the result used by this calculator:
This is a well-known result from stochastic calculus, derived using martingale theory and the optional stopping theorem. A closely related fixed-stakes risk-of-ruin approximation is discussed in Bill Chen and Jerrod Ankenman’s The Mathematics of Poker, especially in the chapters on risk of ruin and uncertain win rates.
📈 Bankroll Random Walk Simulation
Visualizing Brownian motion with drift — why variance ruins winning players
Key Assumptions
- Positive edge: win rate must be greater than zero, otherwise risk of ruin approaches 100%.
- Fixed stakes: the player does not resize or move down as bankroll changes.
- Stable variance: standard deviation is treated as constant throughout play.
- Stable win rate: the model does not account for uncertainty in your true edge.
- Infinite horizon: the estimate is for eventual ruin, not ruin within a fixed number of hands.
- Continuous outcomes: real poker results are discrete; the continuous model slightly overstates risk.
What This Model Does Not Capture
The fixed-stakes approximation is useful as a benchmark, but it is a simplified model and does not fully describe a real poker career.
- Moving down in stakes: the model assumes you stay at the same level forever. A disciplined move-down policy materially reduces real-world ruin risk.
- Uncertain win rate: most players do not know their true edge precisely, especially over small samples. Treating an observed win rate as the true one is optimistic.
- Game quality changes: your edge can shrink as lineups, rake, formats, or table selection change over time.
- Tilt and withdrawals: the formula assumes stable play and no external bankroll drains.
- Discrete poker outcomes: the model is continuous; real results come in discrete hands and sessions.
- Tournament variance: the formula was designed for cash games. Tournament standard deviations are several times higher and the payout structure adds skew that this model does not capture.
For a more realistic bankroll review, combine this calculator with a poker variance simulator and conservative assumptions about your actual win rate.
Related Calculators
- Poker Variance Calculator — visualize downswings and confidence intervals over many hands.
- Kelly Criterion Calculator — compare fixed-stakes bankroll logic with growth-optimal staking.
- Tournament Bankroll Calculator — bankroll planning for MTT players.
- ICM Calculator — estimate tournament equity from stack sizes and payouts.
References and Further Reading
Poker-specific
- Chen, Bill & Ankenman, Jerrod. The Mathematics of Poker. ConJelCo LLC, 2006.
ISBN: 978-1886070257
Discusses poker risk of ruin, bankroll requirements, and uncertainty around win-rate estimates. Useful background for understanding fixed-stakes bankroll models. - Malmuth, Mason. Gambling Theory and Other Topics (Expanded Edition). Two Plus Two Publishing.
ISBN: 978-1880685709 · Related bankroll-management literature; not cited here as the source of the exp(-2μB/σ²) expression.
A foundational poker and gambling text discussing expectation, standard deviation, bankroll requirements, and statistical thinking in gambling decisions.
Mathematical Background
- Feller, William. An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley, 1971.
ISBN: 978-0471257097
Classic textbook containing the mathematical derivation of ruin probabilities for random walks with drift (Chapter XIV). - Ross, Sheldon M. Stochastic Processes. Wiley, 2nd edition, 1996.
ISBN: 978-0471120629
Graduate-level treatment of Brownian motion with drift and the gambler’s ruin problem in continuous time.
Online Resources
- Wikipedia: Gambler’s Ruin — overview of the classical problem.
- Wikipedia: Wiener Process (Brownian Motion) — background on the underlying stochastic process.
Note: The formula RoR = exp(-2μB/σ²) assumes fixed stakes, independent trials, constant parameters, and an infinite time horizon. For a detailed discussion of these assumptions and their practical implications, see Chen & Ankenman (2006), Chapter 22.
Frequently Asked Questions
What formula does this poker risk of ruin calculator use?
It uses the fixed-stakes exponential approximation RoR = exp(-2μB/σ²), where μ is win rate, B is bankroll, and σ² is variance. The output is an estimated probability of going broke under fixed-stakes assumptions.
Is exp(-2μB/σ²) the Mason Malmuth formula?
No. A previous version of this page described the expression that way. The author clarified that this is not the formula presented in his book Gambling Theory and Other Topics, and the attribution has been removed. The expression is best described as a fixed-stakes exponential approximation derived from a continuous random-walk model.
Why is win rate described as drift?
In stochastic-process terminology, drift is the average direction of movement over time. In poker, your win rate is the upward drift, while variance creates the swings around that trend. The vocabulary comes from the underlying random-walk model.
Does this formula apply to sports betting?
Only with caution. The same mathematical structure can be used for fixed-unit betting if you have a reliable estimate of average profit per bet and standard deviation per bet. It is not suitable for variable stakes, parlays, Kelly resizing, or poorly estimated edges.
How do I calculate the ruin probability in Excel or Google Sheets?
Use =EXP(-2*A1*B1/(C1^2)), where A1 is win rate (bb/100), B1 is bankroll (in big blinds), and C1 is standard deviation (bb/100). The result is a decimal between 0 and 1; multiply by 100 to read it as a percentage. For example, 0.1353 means a 13.53% probability of going broke.
What bankroll do I need for 1% risk of ruin?
It depends on your win rate and variance. For a typical winning NLHE player (5 bb/100 win rate, 100 bb/100 standard deviation), the fixed-stakes approximation gives about 4,605 big blinds, or roughly 46 full 100bb buy-ins, for an estimated 1% risk of ruin. See the tables above for other combinations.
Does this formula account for moving down in stakes?
No. The model assumes fixed stakes throughout play. In practice, moving down in stakes when running bad is an effective risk-mitigation strategy that this formula does not capture. With a disciplined move-down policy, your real-world risk of ruin is lower than the formula suggests.
Why does the formula use variance (σ²) instead of standard deviation (σ)?
The formula naturally uses variance because it falls out of the mathematics of the continuous random-walk model. Poker tracking software typically reports standard deviation (σ), so you need to square that value to get variance. For example, if your standard deviation is 100 bb/100, your variance is 100² = 10,000.