Poker Bankruptcy Probability Calculation and Risk Management Model

Context: STRATEGY article: poker-bankroll-probability-risk-management-mqbjvd88

Tool Purpose

The ruin probability calculator is used to evaluate the probability that a player will lose all their bankroll within a given number of hands, based on bankroll size, win rate, standard deviation, and bet size per hand. It is an essential tool for bankroll management, helping players determine appropriate buy-in amounts and game levels for long-term profitability.

Formula Principle

The approximate ruin probability formula is based on a random walk model, assuming independent outcomes and a fixed win rate. The most commonly used formula is:

$$ P(\text{ruin}) \approx \left( \frac{1 - \text{win rate}}{\text{win rate}} \right)^{\frac{\text{bankroll}}{\text{bet per hand}}} $$

Here, the win rate must account for rake. More precise models use standard deviation and win rate, calculated via normal distribution or simulation.

Risk models typically recommend the following rules of thumb:

• To keep the ruin probability below 1%, the bankroll should be at least 300 times the bet per hand (for tournaments) or 100 times (for cash games).
• Adjust based on win rate and variance; for example, players with a high win rate can use a lower multiple.

Usage Steps

1. Determine win rate: Estimate your win rate when profitable (e.g., 60%) based on historical records or opponent skill, adjusting for rake to get net win rate.
2. Calculate standard deviation: For cash games, approximately 100 big blinds per 100 hands; for tournaments, estimate based on buy-in and payout structure.
3. Set acceptable ruin probability: Typically 1% or 5%.
4. Plug into formula or use a calculator: For example, use an online ruin probability calculator to input parameters.
5. Adjust buy-in amount: If the ruin probability is too high, reduce the buy-in or increase the bankroll.

Practical Example

Example: You are playing $1/$2 cash game with a win rate of 55% (after rake) and a standard deviation of $30 per hand (about 15 big blinds). You have a bankroll of $2000. What is the ruin probability after playing 1000 hands?

Solution:

• Bet per hand? In cash games, typically measured by buy-in. Assume a full buy-in of $200 (100 big blinds).
• Simplified formula: Win rate 0.55, bet per hand $200, bankroll $2000, bankroll/bet = 10.
• Ruin probability ≈ ((1 - 0.55) / 0.55)^10 = (0.45 / 0.55)^10 ≈ (0.81818)^10 ≈ 0.134 (13.4%).
• This probability is high (>1%), so it is recommended to reduce the buy-in to $100 (i.e., 50 big blinds). Then bankroll/bet = 20, ruin probability ≈ 0.81818^20 ≈ 0.018 (1.8%), which is acceptable. Alternatively, increase the bankroll to $4000.

Note: The standard win-rate ruin formula applies only to single bets. Actual cash game variance is more complex. Using a professional calculator yields more accurate results.

Frequently Asked Questions

Q1: How is "win rate" defined in the ruin probability formula? A: It refers to the probability of winning each game, after deducting rake. For example, in poker, if rake is 5%, the actual win rate should be reduced accordingly.

Q2: Why is my actual ruin probability higher than the formula suggests? A: The formula assumes a fixed win rate and independent hands, but factors such as fluctuating player skill and withdrawals increase risk. It is recommended to use conservative estimates.

Q3: How does bankroll management differ between tournaments and cash games? A: Tournaments have higher variance and typically require at least 500 buy-ins; cash games can be lower, around 100 buy-ins.

Q4: How can I quickly estimate the required bankroll? A: Use the "Kelly criterion" to optimize: bet proportion = (win rate - loss probability) / odds. However, in poker, half-Kelly or lower is often used.

Further Study

• Learn about the "Kelly criterion" in poker to understand optimal betting proportions.
• Study "Value at Risk (VaR) models" to assess extreme scenarios.
• Use free online calculators (e.g., Poker Bankroll Calculator) to run simulations.
• Read books on poker bankroll management, such as The Mathematics of Poker.