A 75-chip pure bluff into 100 needs more than 42.86% folds to make money in a simple bet-or-fold model. A larger bet needs more folds, even if it tells a convincing story.
Separate the money at risk from the reward
For this model, assume a heads-up river, no rake, no raises, and a hand that always loses when called. When the opponent folds, you win the 100 chips already in the pot. Your own bet comes back to you; it is not additional profit. When called, you lose the 75 chips you just bet.
If F is the probability of a fold, the expected value of betting is F × 100 − (1 − F) × 75. Setting that expression to zero gives the break-even frequency. Divide the bet by the pot plus the bet: 75 ÷ 175 = 42.86%. At exactly that frequency the bluff breaks even; above it the bluff earns chips under the assumptions.
Do not confuse this with the opponent’s price to call. Their 75-chip call would make a final pot of 250, so they need 30% equity. The bluff threshold and the call threshold answer different questions and use different denominators.
A river bluff with no showdown equity
Bluff EV = fold probability × pot − call probability × bet
Break-even folds = bet ÷ (pot + bet)
75 ÷ (100 + 75) = 42.86%
Compare sizes without assuming the same response
A smaller bluff is cheaper, but the opponent may call it more often. A larger bluff risks more and must create enough additional folds to pay for that risk. The formula cannot tell you the response to either size: that requires a defensible range and behavioural estimate.
Suppose a 50-chip bet into 100 gets 40% folds. Its EV is 0.40 × 100 − 0.60 × 50 = +10 chips. Suppose a 100-chip bet gets 55% folds. Its EV is 0.55 × 100 − 0.45 × 100 = +10 chips too. The larger bet succeeds more often, but earns no more in this example. If that larger size gets only 45% folds, its EV falls to −10 chips.
Those fold frequencies are illustrative assumptions, not population reads. For a hand review, calculate several plausible responses rather than choosing a single optimistic number.
| Bet as fraction of pot | Break-even folds |
|---|---|
| One-quarter | 20% |
| One-half | 33.33% |
| Three-quarters | 42.86% |
| Full pot | 50% |
| Twice the pot | 66.67% |
Use blockers to refine a range, not invent one
A blocker is a card in your hand that makes some opposing combinations impossible. It is useful only when those combinations matter to the decision. Holding a card that removes likely calls can help a bluff; holding a card that removes likely folds can hurt it.
On a river with three spades on the board, holding A♠ makes an opponent’s ace-high flush impossible. That does not mean they lack other flushes, strong pairs or calls. Conversely, holding cards from a missed draw can remove hands you hoped they would fold. Start with the action that brought them to the river, then remove combinations using your actual cards.
Write two lists: hands that reach the river and call this size, and hands that reach the river and fold. Weight hands by how often you think they take the earlier actions. A theoretical combination count is not a frequency estimate if you assume every hand reaches the river equally often.
Know when the simple model stops answering the question
On earlier streets, a called bluff can improve, face a raise or require another investment. Those branches need a larger decision tree. Even on the river, a weak hand might sometimes win if checked. A positive bluff EV is not enough if checking has higher EV.
In tournaments, earning chips and improving expected prize money are not interchangeable near meaningful payout pressure. Do not apply this chip-only threshold as a complete tournament recommendation. Use it as an arithmetic check inside a fuller review.
Review the decision before revealing the result. A bluff that got called can still have been reasonable under an accurate model; a successful bluff can still have risked too much against a range unlikely to fold.
Pause and make your decision.
You risk 40 chips to bluff into an 80-chip pot. You estimate 30% folds. Is the bluff profitable in the pure-bluff model?
For your next study session: Take one river bluff from your notes. Calculate its threshold and list the specific calling and folding hands before looking at the result.
Put the idea to work.
Practise six related decisions with worked answers and progress saved in this browser.
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