Understanding a pure river bluff
Content by Tools of Poker Methodology and calculation assumptions
What is this calculation for?
A hand with no showdown value can still make a profitable bluff if the opponent folds often enough. This calculation helps decide whether betting earns more on average than checking.
- Minimum fold frequency: how often the opponent must fold for the proposed bet to break even.
- Expected value (EV): how much the bet gains or loses on average with the estimated response frequency.
In this model checking has zero EV because Hero always loses at showdown and adds no more chips. A positive betting EV favours betting, a negative EV favours checking, and zero leaves both actions equal.
Range analysis estimates how often the opponent folds by counting the hands that can still exist. Hero’s cards and the board remove combos; removing calls raises the fold share, while removing folds lowers it. This helps compare how profitable the same bluff is with different holdings.
The decision and the pure-bluff model
We are heads-up on the river. Villain checks and Hero considers a bet. There are no cards left, raises, rake or side pots. Hero has no showdown value: a call always wins against Hero. On a fold, Hero wins the existing pot; on a call, Hero loses the new bet.
The two exercises use the same risk/reward calculation. The first supplies the fold frequency; the second obtains it from equally weighted available combos after blockers. In real play, the range and responses are estimates; changing those assumptions can change the EV.
A worked example: gain, loss and average result
Current pot: 100 BB · Our bet: 75 BB
Opponent folds: 40% · Opponent calls: 60%
If the opponent folds, we win the existing 100 BB pot. If the opponent calls, we lose our new 75 BB bet. These are the two possible outcomes of the pure bluff.
1. Average gain from folds
Average gain from folds = Fold probability × Current pot
Convert the percentage to a decimal before multiplying.
0.40 × 100 BB = 40.00 BB
2. Average loss from calls
Average loss from calls = Call probability × Our bet
0.60 × 75 BB = 45.00 BB
3. Average result (EV)
Average result (EV) = Average gain from folds − Average loss from calls
This bet loses 5 BB per attempt on average under the stated assumptions. An individual attempt may still win or lose; EV describes the average result.
A separate question: how often must the opponent fold?
The minimum fold frequency balances the risk of our bet against the existing pot we can win. Equality gives zero EV; a fold probability above the exact threshold gives positive EV.
The threshold is about 42.86 percent, but the opponent is estimated to fold only 40 percent. This explains why the same bluff has a negative average result.
| Opponent folds | Average result (EV) | Decision in this model |
|---|---|---|
| 40% | −5.00 BB | Prefer checking |
| 60% | +30.00 BB | Prefer betting |
From blockers and combos to fold frequency
Solved example: Hero has 4s 3s on Ah Kd Qc 9h 2d. The assigned range is JJ, TT, 44, AKs, AQs and KQs. JJ, TT and 44 fold; the other classes call. Every available opponent holding beats Hero.
Current pot: 100 BB · Our bet: 75 BB
1. Remaining combos
Remaining combos = Original combos − Blocked combos
The blocker 4s removes three of the six original 44 combos. The board also blocks suited calls. Count the actual combinations; averaging the six hand classes would give the wrong fold frequency.
2. Folding combos
JJ · TT · 44
3. Opponent folds (%)
Opponent folds = Folding combos / Remaining combos × 100
Keep the exact combo fraction until the final result. For this example, use 15 / 21 for fold probability and 6 / 21 for call probability.
4. Average result (EV)
Average gain from folds = Fold probability × Current pot
Average loss from calls = Call probability × Our bet
Average result (EV) = Average gain from folds − Average loss from calls
Assigned river range
Every remaining physical combo has the same weight. Apply Hero and board blockers before counting folds. These responses apply to the proposed bet size.
JJ · TT · 44
AKs · AQs · KQs
How to practise this exercise
River bluff with fold frequency
The pot, bet and fold/call probabilities are visible from the start at every difficulty. Beginner/easy guide minimum folds, average gain, average loss and EV. Intermediate asks for minimum folds and EV; difficult/nightmare ask only for EV.
River bluff with range analysis
Beginner/easy guide remaining combos, folding combos, fold frequency, the minimum and EV. Intermediate asks for fold frequency, the minimum and EV. Difficult and nightmare ask only for EV. Every difficulty analyses a range; replay keeps you in this exercise.
The timer covers the whole round: 240, 210, 180, 120 and 90 seconds. There are three choices in beginner/easy and four in the other difficulties. Results retain the table and explain every calculation using the actual hand’s inputs.