Confidence Interval

Overview

In poker, a confidence interval (CI) is used to quantify the uncertainty around a player's observed win rate (e.g., big blinds per 100 hands) due to the inherent variance of the game. It provides a range of plausible values for the player's true long-term win rate, given the sample of hands played. For example, a 95% confidence interval of [5 BB/100, 15 BB/100] means that if the same player played many samples of the same size, 95% of those intervals would contain the true win rate—not that there is a 95% chance the true rate is in that specific interval (a common misinterpretation).

Application in Poker

Confidence intervals help players and analysts assess whether a win rate is statistically significant or likely just luck. A wide interval indicates high uncertainty (small sample size or high variance), while a narrow interval suggests more reliable estimates. Typically, a 90% or 95% confidence level is used.

Calculation Example

To compute a CI for win rate, you need:

• Observed win rate (W)
• Standard deviation per 100 hands (SD, often ~100 BB/100 for No-Limit Hold'em)
• Number of hands (N)

The standard error (SE) is SD / sqrt(N/100). Then the CI is W ± z * SE, where z is the z-score (e.g., 1.96 for 95% confidence).

Example: A player wins 10 BB/100 over 10,000 hands with SD = 100 BB/100. SE = 100 / sqrt(100) = 10. 95% CI = 10 ± 1.96*10 = [-9.6, 29.6] BB/100. This wide interval shows the true rate could be negative or highly positive, highlighting the need for larger samples.

Limitations

• Confidence intervals assume normal distribution, which may not hold for small samples.
• They do not account for changes in skill, game conditions, or opponent pool.
• Intervals are only as reliable as the input standard deviation; using an accurate, game-specific SD is crucial.

Related Terms

• Win Rate: The average profit per unit (e.g., BB/100).
• Standard Deviation: A measure of variability in results; higher SD widens intervals.
• Sample Size: Number of hands played; larger samples shrink intervals.
• Variance: The dispersion of results; confidence intervals quantify its impact.
• Statistical Significance: A win rate is significant if the CI does not include zero.

Summary

Confidence intervals provide a realistic range for a poker player's true skill level, helping to separate luck from ability. While not a perfect tool, they are essential for evidence-based bankroll management and performance evaluation.