For professional advantage players and poker grinders, bankroll management is not a guessing game — it is physics. The probability of a bankroll reaching zero before it reaches infinity can be modeled using the mathematics of Brownian motion with drift.
This calculator implements the precise exponential risk of ruin formula popularized by Mason Malmuth in his seminal work “Gambling Theory and Other Topics” and further developed by Bill Chen and Jerrod Ankenman in “The Mathematics of Poker”. The formula is also widely discussed on poker strategy forums like TwoPlusTwo.
Scientific Risk of Ruin Calculator
The Exponential Risk of Ruin Formula: exp(-2μB/σ²)
Unlike simplified calculators that assume coin-flip scenarios, this tool treats your bankroll as a continuous stochastic process. The Risk of Ruin (RoR) formula calculates the exact probability of going broke:
Alternative Notations
You may encounter this formula written in different ways across poker literature and forums:
- Exponential notation: RoR = exp(-2 × winrate × bankroll / variance)
- With standard deviation: RoR = exp(-2 × μ × B / σ²)
- Malmuth notation: RoR = e-2μB/σ²
- Chen/Ankenman notation: RoR = exp(-2WB/V) where W = winrate, V = variance
- Forum shorthand: RoR = exp(-2 * winrate * bankroll / stdev^2)
All of these represent the same mathematical relationship.
Defining the Variables
| Symbol | Name | Poker Term | Typical Unit |
|---|---|---|---|
| μ (mu) | Drift / Expected Value | Win Rate | bb/100 hands |
| σ (sigma) | Standard Deviation | Volatility | bb/100 hands |
| σ² | Variance | Std Dev squared | (bb/100)² |
| B | Barrier / Starting Capital | Bankroll | Big blinds |
| RoR | Risk of Ruin | Bust probability | 0 to 1 (or %) |
Important: If μ ≤ 0 (you are a losing or break-even player), the Risk of Ruin is 100%. This formula only applies to players with a positive edge.
Copy-Paste Formulas for Spreadsheets
Use these ready-made formulas in your bankroll tracking spreadsheet:
=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 in A1 (5), Bankroll in B1 (2000), and Std Dev in C1 (100), the formula =EXP(-2*A1*B1/(C1^2)) returns 0.1353 (13.53% risk of ruin).
The History of the Risk of Ruin Formula
Mason Malmuth & “Gambling Theory and Other Topics”
The exponential risk of ruin formula was popularized in the poker and gambling community by Mason Malmuth in his influential book “Gambling Theory and Other Topics” (first published in 1987, with multiple updated editions since).
Malmuth, who holds a Master’s degree in Mathematics from Virginia Tech, was one of the first authors to apply rigorous statistical analysis to poker bankroll management. As the founder of Two Plus Two Publishing, he has been instrumental in bringing mathematical rigor to poker strategy.
The formula he presented derives from the mathematics of Brownian motion with positive drift — a concept from stochastic calculus originally developed to model random particle movement in physics, now applied to model the random walk of a gambler’s bankroll over time.
Bill Chen & Jerrod Ankenman: “The Mathematics of Poker”
The formula received further rigorous treatment in “The Mathematics of Poker” (2006) by Bill Chen (PhD Mathematics, UC Berkeley; two-time WSOP bracelet winner) and Jerrod Ankenman (MS Applied Mathematics, Columbia; WSOP bracelet winner).
Chapter 22 of their book, titled “Staying in Action: Risk of Ruin”, provides a complete derivation from first principles using martingale theory. Chapter 23 extends the model to account for uncertain win rates — acknowledging that we never know our true win rate with certainty.
Their work established the formula as the standard reference for serious poker players and is frequently cited in academic discussions of gambling mathematics.
The TwoPlusTwo Connection
Both Mason Malmuth (as owner of Two Plus Two Publishing) and the TwoPlusTwo poker forums have been instrumental in spreading this formula throughout the poker community. It remains the standard reference for bankroll calculations discussed in the TwoPlusTwo Strategy forums.
Example: The Danger of Variance
Let’s analyze a solid winning No-Limit Hold’em player:
- Win Rate (μ): 5 bb/100
- Standard Deviation (σ): 100 bb/100
- Bankroll (B): 2,000 bb (20 buy-ins)
The Calculation:
RoR = exp(-20000 / 10000)
RoR = exp(-2)
RoR = 0.1353 or 13.53%
Result: Despite being a winning player, with a 20 buy-in bankroll, this player has a 13.53% chance of going broke due to variance alone.
To achieve 1% risk of ruin: The player would need approximately 4,605 bb (46 buy-ins).
This demonstrates why professional players maintain larger bankrolls than recreational players might expect — the mathematics of variance demand it.
Risk of Ruin Tables (Quick Reference)
Use these pre-calculated tables to quickly estimate your risk of ruin without running the calculator.
Table 1: Win Rate = 5 bb/100, Standard Deviation = 100 bb/100
| Bankroll (BB) | Buy-ins (100bb) | Risk of Ruin |
|---|---|---|
| 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 = more conservative estimates for micro/small stakes)
| Bankroll (BB) | Buy-ins (100bb) | Risk of Ruin |
|---|---|---|
| 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% Risk of Ruin
| Win Rate | σ = 80 bb/100 | σ = 100 bb/100 | σ = 120 bb/100 |
|---|---|---|---|
| 2 bb/100 | 7,378 BB (74 BI) | 11,513 BB (115 BI) | 16,578 BB (166 BI) |
| 3 bb/100 | 4,919 BB (49 BI) | 7,675 BB (77 BI) | 11,052 BB (111 BI) |
| 5 bb/100 | 2,951 BB (30 BI) | 4,605 BB (46 BI) | 6,631 BB (66 BI) |
| 8 bb/100 | 1,844 BB (18 BI) | 2,878 BB (29 BI) | 4,144 BB (41 BI) |
Note: BI = Buy-ins (100 big blinds). Typical NLHE standard deviation is 80-100 bb/100; PLO can exceed 120-140 bb/100.
Mathematical Derivation: Brownian Motion with Drift
For readers interested in the mathematical foundation, here’s why this formula works.
Modeling Your Bankroll as a Stochastic Process
Your poker bankroll over time can be modeled as a random walk or, in continuous time, as Brownian motion with positive drift. If X(t) represents your bankroll at time t:
Where:
- μ = drift coefficient (your win rate pushing the graph upward)
- σ = volatility coefficient (the random fluctuations from variance)
- dW = Wiener process (mathematical “white noise”)
📈 Bankroll Random Walk Simulation
Visualizing Brownian motion with drift — why variance ruins winning players
The Ruin Probability Solution
For a process starting at B (your bankroll) with an absorbing barrier at 0 (going broke), the probability of ever hitting 0 before reaching infinity is:
This is a well-known result from stochastic calculus, proven using martingale theory and the optional stopping theorem.
Key Assumptions of the Formula
- Fixed stakes: You never change your bet size relative to blinds
- Independent trials: Each hand/session is statistically independent
- Constant parameters: Your win rate and variance don’t change over time
- Infinite time horizon: We’re calculating “ever” going broke, not within N hands
- Continuous outcomes: The formula slightly overestimates risk because poker has discrete results
Despite these idealizations, the formula provides an excellent approximation for practical bankroll management decisions.
Related Calculators
Combine this Risk of Ruin calculator with other bankroll management tools:
- Kelly Criterion Calculator — Calculate optimal bet sizing for maximum growth
- Poker Variance Calculator — Visualize variance and confidence intervals over many hands
- Tournament Bankroll Calculator — Risk of Ruin calculations for MTT players
- ICM Calculator — Independent Chip Model for tournament equity
References & Further Reading
The exponential risk of ruin formula presented in this calculator is derived from well-established mathematical theory. Below are the primary sources:
Primary Sources
- Malmuth, Mason. Gambling Theory and Other Topics. Two Plus Two Publishing, 1987 (5th edition 2004).
ISBN: 978-1880685419 · Publisher’s page
The foundational text that popularized the exponential RoR formula in the poker community. Chapters on bankroll requirements and risk of ruin remain the standard reference. - Chen, Bill & Ankenman, Jerrod. The Mathematics of Poker. ConJelCo LLC, 2006.
ISBN: 978-1886070257 · Amazon
Chapter 22 (“Staying in Action: Risk of Ruin”) provides a rigorous derivation from first principles. Chapter 23 extends the model to uncertain win rates.
Mathematical Background
- Feller, William. An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley, 1971.
ISBN: 978-0471257097
The 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
- TwoPlusTwo Poker Theory Forum — Active discussion of bankroll management and RoR calculations
- Wikipedia: Gambler’s Ruin — Mathematical overview of the classical problem
- Wikipedia: Wiener Process (Brownian Motion) — Background on the stochastic process underlying the formula
Note: The formula RoR = e−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 (FAQ)
What is the Mason Malmuth Risk of Ruin formula?
The Mason Malmuth Risk of Ruin formula is RoR = exp(-2μB/σ²), where μ is your win rate, B is your bankroll, and σ² is variance (standard deviation squared). This exponential formula was popularized by Mason Malmuth in his book “Gambling Theory and Other Topics” and calculates the probability of going broke before your bankroll grows infinitely, assuming fixed stakes play.
Why is my Winrate labeled as “Drift”?
In stochastic calculus, “drift” represents the average trend of a process over time. In gambling and poker, your Win Rate is the drift — the force pushing your bankroll graph upwards over time. Variance (sometimes called “diffusion”) creates the jagged ups and downs around this trend. This terminology comes directly from Brownian motion mathematics.
Does this formula apply to sports betting?
Yes, provided you can standardize your units. If you use a fixed stake size, you can calculate your average profit per bet (μ) and the standard deviation per bet (σ) to find the required bankroll (B) in betting units. The mathematics are identical — only the units change.
How do I calculate Risk of Ruin in Excel or Google Sheets?
Use this formula: =EXP(-2 * A1 * B1 / (C1^2)) where A1 contains your win rate (bb/100), B1 contains your bankroll (in big blinds), and C1 contains your standard deviation (bb/100). The result will be a decimal between 0 and 1 — multiply by 100 to get the percentage. For example, 0.1353 means 13.53% risk of ruin.
What is the “Kelly” connection?
There is a direct mathematical link between Risk of Ruin and the Kelly Criterion. If you bet a full Kelly fraction, your Risk of Ruin to reach a half-bankroll is technically 50%. However, with continuous Kelly resizing, true ruin (reaching exactly $0) is theoretically impossible. The exponential RoR formula assumes fixed stakes, which is how most poker players actually play — you don’t resize your $1/$2 buy-in based on every session result.
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), you need approximately 4,605 big blinds or 46 buy-ins for a 1% risk of ruin. See the tables above for different win rate and variance combinations.
Is the formula in “The Mathematics of Poker” the same as Malmuth’s?
Yes. Bill Chen and Jerrod Ankenman present the same exponential formula in Chapter 22 of “The Mathematics of Poker” (2006). They derive it rigorously from first principles using Brownian motion theory. Chapter 23 extends the model to include uncertain win rates — accounting for the fact that we never know our true win rate with certainty.
Why does the formula use variance (σ²) instead of standard deviation (σ)?
The formula naturally uses variance because it emerges from the mathematical derivation of Brownian motion. However, poker tracking software typically reports standard deviation (σ), so you need to square this value to get variance (σ²). For example, if your standard deviation is 100 bb/100, your variance is 100² = 10,000.
Where does the risk of ruin formula come from?
The formula RoR = e−2μB/σ² derives from the mathematics of Brownian motion with positive drift. It was popularized in the gambling community by Mason Malmuth in “Gambling Theory and Other Topics” (1987) and later given rigorous treatment by Bill Chen and Jerrod Ankenman in “The Mathematics of Poker” (2006). The underlying mathematics date back to William Feller’s foundational work on random walks in the 1950s-60s.
Does this formula account for moving down in stakes?
No. This formula assumes fixed stakes — you play the same level regardless of your bankroll. In practice, moving down in stakes when running bad is an effective risk mitigation strategy that the formula doesn’t capture. If you have a disciplined move-down policy, your actual risk of ruin is lower than the formula suggests.