How often a bluff has to work

A pure bluff risks your bet to win the pot. When your opponent folds, you win the pot. When they call, you lose the bet. The bluff breaks even when:

With 100 in the pot and a bet of 50:

If your opponent folds more than a third of the time, the bluff makes money. This is the bluff break-even, and it is the same sum the trainer's math page gives.

The contribution to a bet's value from making opponents fold is its fold equity. A pure bluff depends entirely on fold equity because it never wins at showdown.

Small bets need few folds. A third-pot bluff needs one fold in four, but a small bet also gives the caller a good price, so expect fewer folds. Big bets need many folds: a pot-sized bluff has to work half the time, and twice the pot needs two folds in three.

The same bet gives two different numbers, one for each player.

The bettor's number is the folds needed above: 33.3% for a half-pot bet. The caller's number is the price of the call, their required equity: they call 50 to play for 200, so 25%.

They differ because the two players risk different amounts against different pots. The bluffer risks 50 to win 100. The caller risks 50 to win 150, since the bluffer's 50 is in the pot by then. The caller gets the better price.

Subtract the bettor's required fold frequency from 100% and you get the caller's minimum defense frequency: against a half-pot bet, the caller has to continue with at least 66.7% of their range, or any two cards make a profitable bluff. That is a baseline for an opponent who bluffs well. Against one who rarely bluffs, folding more is right.

The third column of the table answers a different question: if you bet a range of strong hands and bluffs, what share of it can be bluffs?

A caller holding a hand that only beats a bluff wins when you bluff and loses when you value bet. Their call breaks even when your bluffs make up exactly their required equity. So against a half-pot bet the balanced range is 25% bluffs, one bluff for every three value hands. Against a pot-sized bet it is a third, one bluff for every two value hands.

Bet more of your bluffs than that and calling every bluff-catcher makes money against you. Bet fewer and your opponent can fold those hands and lose nothing. The bluff-to-value ratio is the same idea as a ratio.

Bigger bets carry more bluffs: a bigger bet gives the caller a worse price, so you can bluff more before calling becomes profitable.

All of this is for a pure bluff, a hand that never wins when called. A hand with outs is a semi-bluff: when it is called it can still hit, so it needs fewer folds. A flush draw on the flop hits about a third of the time when it sees two more cards, so it can need far fewer folds than the table shows.

On the river there are no more cards, so a bluff cannot improve. Real river bluffing hands can still beat some calls, and blockers change the available combinations. The table applies to the polarised model described above, not every possible betting range.