A Practical Guide to Poker Bankruptcy Probability Calculation and Risk Management Models

Tool Purpose

Bankroll probability calculation is a core tool for poker bankroll management, used to assess the likelihood of a player losing all their capital given a certain bankroll, win rate, and variance. By quantifying risk, players can set appropriate stakes, bet sizes, and bankroll growth strategies to avoid "going broke" due to short-term variance.

Calculation Formula Principles

Kelly Criterion

The Kelly Criterion is used to calculate the optimal bet size to maximize long-term capital growth. For poker, it is typically simplified to:

$$f = \frac{bp - q}{b}$$

where:

• $f$ = recommended fraction of bankroll to bet
• $b$ = odds (ratio of pot to bet, e.g., if you win $n$ for every $1$ invested, then $b=n$)
• $p$ = probability of winning
• $q$ = 1 - p (probability of losing)

In practice, poker bets do not have fixed odds, so "fractional Kelly" is often used to reduce risk.

Bankroll Probability under Fixed Win Rate (Normal Approximation)

For independent and identically distributed hands, using the normal distribution approximation:

$$P_{\text{bankrupt}} \approx \Phi\left( \frac{-B \cdot \mu / \sigma}{\sqrt{t}} \right)$$

where:

• $B$ = bankroll (in BB)
• $\mu$ = expected win per hand (e.g., +5 BB/100 hands)
• $\sigma$ = standard deviation per hand (e.g., 80 BB/100 hands)
• $t$ = number of hands

If time is considered infinite, the bankroll probability can be simplified to:

$$P_{\text{bankrupt}} = \left( \frac{1 - s}{1 + s} \right)^{B / \sigma}$$

where $s = \mu / \sigma$ (Sharpe ratio).

How to Use

1. Estimate Win Rate and Variance: Based on historical data or industry consensus, determine your average win rate (e.g., 5 BB/100 hands) and standard deviation (e.g., 80 BB/100 hands).
2. Set Acceptable Bankrupt Probability: Usually 1%–5%.
3. Plug into Formula to Calculate Minimum Bankroll: Use the normal approximation or the more precise bankruptcy formula.
4. Design Bankroll Move-up Rules: For example, move up when bankroll reaches 20x the current stake, move down when it falls below 30x.
5. Adjust Dynamically: Recalculate as win rate changes.

Practical Example

Assume you play at NL100 cash tables (big blind $1), with an average win rate of 5 BB/100 hands and standard deviation of 80 BB/100 hands.

• Target bankrupt probability < 1% (0.01).
• Use the formula: $P = \left( \frac{1 - s}{1 + s} \right)^{B / \sigma}$, where $s = 5/80 = 0.0625$.
• Take logarithms to solve: $\ln(0.01) = \frac{B}{80} \cdot \ln\left( \frac{1-0.0625}{1+0.0625} \right) \approx \frac{B}{80} \cdot \ln(0.8824)$.
• $\ln(0.01) = -4.605$, $\ln(0.8824) = -0.125$, so $B = 80 \times (-4.605)/(-0.125) \approx 2947$ BB.
• That means a minimum bankroll of about 2947 BB = $2947.
• If your bankroll is only $1000, the bankrupt probability is roughly $\left(0.8824\right)^{1000/80} = (0.8824)^{12.5} \approx 0.26$, or 26%, which is too risky.

Common Questions

• Question: My win rate is unstable, how do I use the formula?
  Answer: Use conservative estimates (e.g., lowest expected win rate) or run multiple simulations. You can also use Monte Carlo simulation tools.
• Question: Can the Kelly Criterion be applied directly to bet sizing?
  Answer: In tournaments or cash game betting decisions, the Kelly Criterion can guide value bet sizing, but you must account for opponent responses. "Fractional Kelly" (e.g., 1/4) is recommended to reduce risk.
• Question: Does the bankrupt probability formula apply to tournaments?
  Answer: Tournament structures are more complex, involving ICM and probability of making the money. Use specialized tournament bankroll models, e.g., bankroll ≥ 40 buy-ins.

Further Learning

• Study ICM (Independent Chip Model) and its impact on tournament bankrolls.
• Read articles such as "Poker Bankroll Management: Quantifying Risk."
• Use online bankrupt probability calculators (e.g., Poker Bankroll Calculator) for scenario simulations.