Part 2 · Lesson 6 of 25 · Free
The tells are in the data
A player you have never met raises 6 of the first 20 hands: 30%. That number feels like a read, but it fits a true raise rate anywhere from about 10% to 50%. A statistic is an estimate with an error bar, and over a small sample the error bar is wide.
The hand
Three players could be behind those 6 raises in 20 hands: a Tight-aggressive player who raises 20% of hands, a Loose-aggressive one who raises 30%, or a Maniac who raises 45%.
Predict first
After 20 hands, which of the three can you rule out?
- None of them
- The Tight-aggressive player
- The Maniac
Show the answer
Answer: None of them
6 of 20 fits any true rate from about 10% to 50%, so all three are still plausible. Twenty hands are a hint, not a read.
Working it out
The error bar for a frequency is about 1.96 × √(p × (1 − p) ÷ n) on each side of what you saw. Here p = 6 ÷ 20 = 0.30 and n = 20: 1.96 × √(0.30 × 0.70 ÷ 20) ≈ 0.20, so from 10% to 50%. A more careful method for small samples, the Wilson interval, gives 15% to 52%. Either way, all three styles are still in play.
The error bar narrows only with the square root of the sample: four times the hands for half the uncertainty. Use a small sample as a weak prior, combine it with what each style usually looks like, and update as the evidence grows. At the table here, try hiding the bots' styles and guessing them from their actions.
Formula
Frequency seen p = k ÷ n; 95% error bar ≈ p ± 1.96 × √(p × (1 − p) ÷ n), shrinking like 1 ÷ √n (the Wilson interval is more accurate for small samples)
Check your understanding
He raised 30 of the 100 hands you saw. Where is his true raise rate, with 95% confidence?
- Exactly 30%
- About 21% to 39%
- About 10% to 50%
Show the answer
Answer: About 21% to 39%
1.96 × √(0.30 × 0.70 ÷ 100) ≈ 0.09, so from about 21% to 39%. Five times as many hands as 20 made the error bar √5 ≈ 2.2 times narrower.
To halve the uncertainty of a stat, you need about how many hands?
- Twice as many
- Four times as many
- Ten times as many
Show the answer
Answer: Four times as many
Uncertainty shrinks with the square root of the sample, so halving it needs four times the hands.