Part 1 · Lesson 3 of 25 · Free
Bayes’ theorem in poker: updating hand ranges
On the river, 30% of the Loose-aggressive player's hands are missed draws. Then he bets, and the share of bluffs among his hands drops to 17.6%. Every action is evidence. Bayes' theorem tells you how far it should move your belief, starting from what you believed before: the prior.
The hand
River. You hold second pair, a bluff catcher: it beats his missed draws and nothing else. He bets 15 bb into 30 bb, so your call needs to win 15 ÷ (45 + 15) = 25% of the time.
Predict first
30% of his hands are missed draws. He bets every hand that beats you, but only half of his missed draws. Once he bets, are bluffs more or less than 30% of his hands?
- More than 30%
- Still 30%
- Less than 30%
Show the answer
Answer: Less than 30%
All his strong hands bet and only half his bluffs do, so a bet comes from strength more often than before: bluffs fall to 17.6%.
Working it out
Picture 100 of his hands: 30 missed draws and 70 that beat you. All 70 strong hands bet, and 15 of the missed draws do. Of the 85 bets, 15 are bluffs: 15 ÷ 85 = 17.6%. Your call needs at least 25%, so you fold, even though he does bluff here.
In practice you keep a range for each player and narrow it with each action. That is this same calculation, repeated hand type by hand type, street after street.
Formula
P(bluff | bet) = P(bluff) × P(bet | bluff) ÷ [P(bluff) × P(bet | bluff) + P(value) × P(bet | value)]
Check your understanding
He holds a bluff 20% of the time and value 80%. He bets all his value hands and half his bluffs. He bets: what is the chance it is a bluff, in %?
Show the answer
Answer: 11.1%
0.2 × 0.5 ÷ (0.2 × 0.5 + 0.8 × 1) = 0.1 ÷ 0.9 ≈ 11.1%.
A Rock holds a bluff 5% of the time and bets every hand he has here, bluffs and value alike. He bets: what is the chance it is a bluff, in %?
Show the answer
Answer: 5%
0.05 × 1 ÷ (0.05 × 1 + 0.95 × 1) = 5%. When every hand bets, the bet carries no information, and the prior stands.