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 utilizes the precise exponential formula often cited in advanced gambling literature (such as “Gambling Theory and Other Topics” by Mason Malmuth) and forums like TwoPlusTwo.
Calculate P(Ruin) based on RoR = e^(-2μB / σ²).
The Mathematics: $e^{-2\mu B / \sigma^2}$
Unlike simplified calculators that assume coin-flip scenarios, this tool treats your bankroll as a continuous fluid path. The formula for the Risk of Ruin (RoR) is:
Defining the Variables
- $\mu$ (Mu) – Winrate: Your expected drift or edge. In poker, this is usually expressed as bb/100 (big blinds per 100 hands). It must be positive; if $\mu \le 0$, the Risk of Ruin is 100%.
- $\sigma$ (Sigma) – Standard Deviation: A measure of variance. In No-Limit Hold’em, this is typically between 80 and 120 bb/100. In PLO, it can exceed 140 bb/100. Note that the formula uses Variance ($\sigma^2$).
- $B$ (Bankroll): Your total starting capital, expressed in the same units as your winrate (e.g., Big Blinds).
Example: The Danger of Variance
Let’s analyze a solid winning poker player:
- Winrate ($\mu$): 5 bb/100.
- Std Dev ($\sigma$): 100 bb/100.
- Bankroll ($B$): 2,000 bb (20 Buy-ins).
The Calculation:
$$RoR = e^{\frac{-2 \times 5 \times 2000}{100^2}} = e^{-2} \approx 0.1353$$
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 reduce this risk to under 1%, the player would need to increase their bankroll to roughly 4,600 bb (46 Buy-ins).
Frequently Asked Questions (FAQ)
Why is my Winrate labeled as “Drift”?
In stochastic calculus, “drift” represents the average trend of a process over time. In gambling, your Winrate is the force pushing your graph upwards, while Variance (diffusion) creates the jagged ups and downs.
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 ($\mu$) and the standard deviation per bet ($\sigma$) to find the required bankroll ($B$) units.
What is the “Kelly” connection?
There is a direct mathematical link. If you bet a full Kelly fraction, your Risk of Ruin is technically high (often cited as 50% for reaching a half-bankroll, but ruin is theoretically impossible with continuous resizing). This formula assumes fixed stakes, which is how most people actually play.