FUNDAMENTALS
Bayesian inference at the poker table
Update a prior range from observed betting behaviour using Bayes’ rule and base rates.
A bet changes a probability
Suppose a player has a strong hand 30% of the time. They bet 80% of strong hands and 20% of weak hands. What should you believe after seeing a bet?
Strong-and-bet outcomes occupy 0.30 × 0.80 = 24% of all trials. Weak-and-bet outcomes occupy 0.70 × 0.20 = 14%. Among bets, the strong proportion is 24 / 38, about 63.158%.
P(S | B) = P(B | S)P(S) / [P(B | S)P(S) + P(B | ¬S)P(¬S)]
Do not confuse the conditional directions
“They bet 80% of strong hands” is P(bet | strong). It is not the probability they are strong given a bet. Reversing a conditional without considering the prior is a base-rate error.
Treat the model as an assumption
The numbers in the drill are given. In a real decision they would be estimated, and a limited sample may produce substantial uncertainty.
Use the calculation to understand the effect of evidence. Do not mistake a precise computed posterior for precise knowledge of the opponent.