Variance & Downswings

📊 What Is Variance in Poker?

Variance describes the natural fluctuation in poker results around your true win rate. Even with a positive expected value, short-term results can deviate dramatically from expectations.

In poker, your results per session are a random variable. Over \(n\) hands with win rate \(\mu\) (bb/100) and standard deviation \(\sigma\) (bb/100):

$$\text{Total Winnings} \sim N\left(\frac{n \cdot \mu}{100},\ \frac{n \cdot \sigma^2}{100}\right)$$

The standard deviation of your total results over \(n\) hands is:

$$\text{SD}_{n} = \sigma \sqrt{\frac{n}{100}}$$

Typical Standard Deviations by Game Type

📏 Confidence Intervals — What Your Results Mean

A confidence interval tells you the range within which your true win rate likely falls, given your observed results.

For a sample of \(n\) hands with observed win rate \(\hat{\mu}\) and standard deviation \(\sigma\):

$$\text{95% CI} = \hat{\mu} \pm 1.96 \times \frac{\sigma}{\sqrt{n/100}}$$

Worked Example

You have played 50,000 hands with an observed win rate of +4 bb/100 and SD of 90 bb/100.

$$\text{SE} = \frac{90}{\sqrt{50{,}000/100}} = \frac{90}{\sqrt{500}} = \frac{90}{22.4} \approx 4.02 \text{ bb/100}$$ $$\text{95% CI} = 4 \pm 1.96 \times 4.02 = 4 \pm 7.88 = [-3.88,\ +11.88] \text{ bb/100}$$

At 50,000 hands, you cannot even confirm you are a winning player at 95% confidence! Your true win rate could be anywhere from –3.9 to +11.9 bb/100.

How Many Hands to Confirm a Win Rate?

To confirm you are a winner (true win rate > 0) at 95% confidence:

$$n = \left(\frac{1.645 \times \sigma}{\hat{\mu}}\right)^2 \times 100$$

📉 Downswing Probability Calculations

How likely is a severe downswing for a winning player? Let us calculate.

The probability of experiencing a downswing of at least \(D\) big blinds at some point over \(n\) hands is approximately:

$$P(\text{max drawdown} \geq D) \approx 2 \cdot \Phi\left(\frac{-D + \mu \cdot n/100}{\sigma \sqrt{n/100}}\right)$$

Where \(\Phi\) is the standard normal CDF. For practical purposes, we can use simulation results:

Downswing Probabilities: 5 bb/100 Winner, SD = 100 bb/100

A 5 bb/100 winner over a career of 100,000 hands has a 52% chance of experiencing a 500 bb downswing at some point. These are not rare events — they are expected.

The "Never Recover" Scenario

The probability of busting a bankroll of \(B\) bb for a player with win rate \(\mu\) and SD \(\sigma\) is approximately:

$$P(\text{ruin}) \approx e^{-2\mu B / \sigma^2}$$

For \(\mu = 5\) bb/100, \(\sigma = 100\) bb/100, and bankroll \(B = 2000\) bb:

$$P(\text{ruin}) \approx e^{-2 \times 0.05 \times 2000 / 100} = e^{-2} \approx 13.5\%$$

🧠 Practical Takeaways

1. Never Evaluate Over Less Than 50,000 Hands

Results over 5,000–20,000 hands contain almost no signal about your true win rate. Use this period to evaluate your process and decision quality, not your results.

2. Expect Downswings — Plan for Them

A 500 bb downswing is a routine event for a winning player. Budget emotionally and financially for downswings of 300–500 bb at 6-max NLHE before they happen.

3. The Required Sample Formula

To estimate your true win rate within ±X bb/100 at 95% confidence:

$$n = \left(\frac{1.96 \times \sigma}{X}\right)^2 \times 100$$

For SD=100 and X=3 bb/100: \(n = (1.96 \times 100 / 3)^2 \times 100 \approx 426{,}000\) hands — far more than most players ever play.

4. Tilt Is Variance's Amplifier

Variance does not cause tilt — your reaction to variance does. Understanding the math that downswings are inevitable and expected removes the false "I must be playing badly" narrative that drives tilt.