Fold equity for a single-street bluff

You bet 50 into a pot of 100. If the opponent folds, you gain the existing 100. If the opponent calls and your bluff loses, you lose the new 50. Those two outcomes are enough to calculate the fold frequency needed by a pure bluff.

The model here has one opponent, no raises, no later betting, no fees and no ties. Your hand always loses when called. The calculator answers what a supplied fold probability implies under those conditions. It cannot discover how often an opponent will actually fold.

Ace: “I have one tray for a fold and one for a call.”

Celeste: “Put a gain of 100 in the fold tray and a loss of 50 in the call tray. Those are our two net outcomes.”

Measure the pot before you bet

Let P be the pot before your bet, B the new amount you risk, and q the opponent’s fold probability. In the formula, 40% is written as 0.40. The desk accepts the percentage directly, so its fold-frequency field takes 40 for that assumption.

P includes chips already committed by every player, including dead money from players who folded. There are no side pots in this exercise, and the bettor can win the whole existing pot. B is only the new bet. Keep both amounts in the same unit.

When the opponent folds, your uncalled bet is returned. If you put 50 into a pot of 100, the chips coming back may total 150, but 50 were yours immediately before this decision. The net gain is 100. Counting the returned bet as profit would overstate the winning outcome.

A stack comparison makes the distinction visible. Start with 500 behind, put out 50, then receive your 50 back along with the existing 100. You finish with 600 behind, a gain of 100. If the opponent calls and wins, you finish with 450, a loss of 50.

Your earlier contributions are already part of P. Losing the hand may feel like losing everything you put in from the start, but this calculation measures the new decision. Do not add earlier bets to B again. A whole-hand profit record and a decision EV calculation start at different moments.

Weight the two outcomes

The fold branch happens with probability q and gains P. The call branch happens with probability 1 – q and loses B. Expected value is their weighted sum: EV = qP – (1 – q)B.

For P = 100, B = 50 and q = 40%, the fold branch contributes 0.40 x 100 = 40. The call branch contributes a loss of 0.60 x 50 = 30. Subtract to get EV = 10 chips per instance of this model.

An EV of 10 does not mean the next hand returns 10. Under the stated model, its result is either +100 or -50. Ten is the probability-weighted average. Actual sequences can contain runs of calls or folds, and no fixed number of attempts guarantees the average.

For a simple counting check, imagine 100 hypothetical instances containing exactly 40 folds and 60 calls. The folds gain 4,000 chips and the calls lose 3,000, leaving 1,000 overall, or 10 per instance. This illustrates the weights; it does not predict a future block of 100 hands.

The extreme inputs also check the formula. At q = 0%, every bluff is called and EV is -B. At q = 100%, every opponent folds and EV is +P. Any intermediate probability must give an EV between those two net outcomes.

Find the break-even fold frequency

Set EV to zero and expand the loss term: qP – B + qB = 0. Collect the q terms to get q(P + B) = B. The break-even fold probability is therefore B/(P + B).

For a 50-chip bet into 100, the exact fraction is 50/150 = 1/3. The percentage repeats: 33.333333…%. More than one-third folds gives positive EV in this model; less gives negative EV. Equality gives zero.

The denominator contains the existing pot plus your risk. Dividing 50 by 100 gives the bet-to-pot ratio, 50%, which answers a different question. The break-even calculation is 50 divided by 150 because both the gain and loss affect the weighted result.

Use the exact fraction when comparing close inputs. With a pot of 100 and a bet of 50, a 33.33% fold frequency gives EV -0.005000. At 33.34%, EV is +0.010000. Neither percentage is exactly one-third. A rounded display should not turn a small loss into a break-even label.

A pot-sized bet offers a boundary you can enter exactly. Set both P and B to 100 and q to 50%. The fold gain and called loss each contribute 50, so EV is 0.000000. Change q to 49% and the result is -2; change it to 51% and the result is +2.

Ace: “The dealer pushed 150 back, so I wrote a profit of 150.”

Celeste: “Separate our returned 50 first. The pot we gained was 100; our own uncalled bet adds no profit.”

Test more than one plausible estimate

The required fold frequency comes from P and B. The actual fold frequency is a separate assumption about the opponent’s response to this bet in this situation. A result above the threshold says the arithmetic favors the supplied estimate; it does not establish that estimate as true.

Keep P at 100 and B at 50, then test 30%, 40% and 50% separately. You get -5, +10 and +25. A plausible range that runs from 30% to 40% crosses the break-even boundary. That uncertainty matters even though the midpoint would produce a positive result.

For this fixed pot and bet, each extra percentage point of folds increases EV by 1.5 chips. The formula can be written q(P + B) – B, so increasing q by 0.01 changes EV by 0.01 x 150. This makes it easy to inspect how sensitive a close result is to your estimate.

A range of 35% to 40% would produce EVs from 2.5 to 10. A range of 25% to 30% would produce -12.5 to -5. Because EV increases steadily with q when P and B are fixed, the endpoints bound the results between them. They still do not prove that the chosen range describes the opponent.

An estimate needs context: the opponent’s possible hands, the board, prior action and the offered size. A fold percentage copied from unrelated situations may describe a different response. Record what observation or assumption supports the number before interpreting the result.

A small observed sample is also different from a known probability. Seeing four folds in ten examples records what happened in those ten cases. It does not establish a 40% future rate, especially if their boards, positions or bet sizes differ.

Compare sizes with their own assumptions

Increasing B raises the fold frequency required for a pure bluff. Holding q fixed also makes each call more costly. The opponent’s response may change with the size, however, so comparing real bets requires a separate response estimate for each.

This table does not identify the best real bet. The 25-chip bet might receive a different response from the 100-chip bet. A size that looks better while q is held fixed may look worse after its own response is considered.

For a different arithmetic exercise, use P = 80 and B = 60. The threshold is 60/140 = 3/7, about 42.857143%. At 40% folds, EV is 32 – 36 = -4. At 50%, it is 40 – 30 = 10.

Change that bet to 20 while keeping the hypothetical 40% response. The threshold becomes 20/100 = 20%, and EV becomes 32 – 12 = 20. You have changed the risk while keeping the response fixed for comparison. You have not shown that a smaller bet gets the same folds.

Keep checking and semi-bluffing separate

The desk evaluates the supplied bet against a zero-result reference. It does not calculate checking. A positive betting EV alone does not prove betting is the best action; choosing between actions requires their values under a consistent model.

For example, if one complete model gives a bet an EV of 2 and checking an EV of 6, checking is better under those assumptions even though the bet is positive. Those checking values need their own evidence. The bluff formula cannot supply them.

In the strict final-street case where your hand always loses if checked to showdown, checking down can have a zero result from the present decision point. Earlier streets or hands with showdown value can behave differently. State the checking assumption instead of assigning zero automatically.

A semi-bluff changes the called branch because the hand can sometimes win after a call. The present model assigns every call a loss of B, so it cannot value that possibility. It also contains no field for a draw, showdown equity or later action.

For a separate called-hand analysis, the relevant win chance is conditional on the opponent calling. That can differ from your chance against all hands the opponent held before deciding. Hands that fold no longer belong to the calling range.

A draw-completion percentage is also different from a showdown win percentage. A completed draw can lose, and some missed draws can still win. The one-card and two-card draw lesson explains that distinction without feeding a draw percentage into this pure-bluff model.

Recognize hands that need a wider model

Multiway pots need more than one opponent response. Winning immediately generally requires every remaining opponent to fold. Two separate 50% fold estimates do not automatically establish a 25% chance that both fold.

Multiplying 0.50 by 0.50 assumes the responses are independent. In a real action sequence, the later opponent may respond differently after seeing a fold or call. Without independence, you need the first opponent’s fold probability and the second opponent’s probability of folding conditional on that first fold.

Raises are excluded here. If an opponent can raise, the action tree has another branch. If you can call that raise or bet again later, the amount at risk may exceed B. The two-outcome desk does not price any of those choices.

Fees and pot deductions can change the amount actually won. Side pots change eligibility, and ties create additional outcomes. Tournament prize considerations also require more than a chip-EV figure. The displayed result applies to the stated one-opponent model without those features.

Ace: “My worksheet says 40%, but the cards do not reveal that frequency.”

Celeste: “Keep 40% labeled as an estimate. Test nearby values and record which situations support it.”

Run the bluff study desk

Start with the default pot 100, bet 50 and estimated folds 40%, then choose Check the bluff model. The receipt shows the exact break-even fraction, a rounded percentage and EV of 10. Read the result as positive under those inputs, rather than as a forecast of the opponent’s next action.

Enter pot and bet amounts above zero through 1,000,000, using at most two decimal places and the same unit. Enter fold frequency from 0 through 100 percent, also with at most two decimals. Use plain numbers without currency signs, thousands separators or scientific notation.

Changing any field clears the old result. Use Check the bluff model after editing so the receipt matches the visible inputs. Reset example restores 100, 50 and 40. Blank fields are missing information, not zero, and the desk rejects a zero pot or zero bet.

The receipt compares the calculation with the exact break-even boundary before rounding its percentage. EV is displayed with up to six decimal places. Those digits describe the arithmetic; they do not add confidence to an estimated fold frequency.

Keep a short study note with the pot before betting, new risk, assumed fold frequency and resulting net outcomes. For the default case: “100 already in the pot; risk 50; assume 40% folds; gain 100 on a fold or lose 50 on a call; EV 10.” For the next review, compare your response assumption with evidence from relevant situations.

Use the desk for off-table review. More worked poker exercises are available in Info & Tools. Preserve the same reference point when comparing calculations so a returned bet, earlier contribution or future wager is not counted in the wrong branch.

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