A hand has value against a range
River bluff catching with top pair
Top pair is a hand category, not a decision. With no cards left to come, the question is how often this exact hand beats the combinations in the opponent’s betting range. Change that range and the answer can change.
How to approach the decision
- Remove the board and your cards from the opponent’s combinations.
- Count which remaining value hands and bluffs your hand beats.
- Compare range equity with the break-even price. Folding has zero additional cost at this decision.
Your practice hand
Villain bets 15 bb into 20 bb on the river.
Your hand
K♣4♦Board
K♥9♠5♣2♦7♥Facing
15 bb into 20 bbVillain range: custom · 50 of 84 combos live
AK,KQ,KJ,K9s,99,55,86s,QJs,JTs,T8s,64sVillain bets better kings, sets, and the 86s straight for value, and missed draws (QJs, JTs, T8s, 64s) as bluffs. With your cards removed that is 50 combos.
Practise this exact hand →Show the worked answer
Your king-four beats every bluff and nothing else, so your equity is simply the bluff share. A 15 bb bet into 20 needs 30%, and 15 bluffs out of 50 combos is exactly 30%: you are indifferent to the chip. One more bluff combo makes it a call, one fewer a fold.
The working
- Range. The stated range is 84 combos. 34 of them use a card you can see (your hand or the board), which leaves 50.
- Equity. You win 30.0% and chop 0.0%, so equity is 30.0% + 0.0% ÷ 2 = 30.0%. The river is dealt, so every combo is evaluated once and the number is exact.
- Price. Calling risks the bet to win the pot plus the bet, so break-even is bet ÷ (pot + 2 × bet) = 15 ÷ (20 + 2 × 15) = 30.0%.
- EV. equity × (pot + bet) − (1 − equity) × bet = 0.3000 × 35 − 0.7000 × 15 = 0.00 bb. Folding is 0, so that is the whole difference between the two options.