The two break-even numbers
Every bet size creates a threshold for the caller and one for the bettor. Both are pot odds, and solvers live between them.
6 minute read
You already know the caller's number: facing a bet, you need equity of at least the call divided by the pot after your call. Turn the same arithmetic around and you get the bettor's number. A bluff risks the bet to win the pot, so it breaks even when it succeeds often enough to cover the risk.
Bluff break-even, alpha
α = bet ÷ (pot + bet)
A pot-size bet must work half the time; a half-pot bet one time in three.
Minimum defence frequency
MDF = 1 − α = pot ÷ (pot + bet)
The share of the range the defender must continue with so that a bluff with no equity is not automatically profitable.
| Bet | Caller needs | Bluff must work | Minimum defence | River value : bluff |
|---|---|---|---|---|
| ¼ pot | 16.7% | 20.0% | 80.0% | 5 : 1 |
| ⅓ pot | 20.0% | 25.0% | 75.0% | 4 : 1 |
| ½ pot | 25.0% | 33.3% | 66.7% | 3 : 1 |
| ⅔ pot | 28.6% | 40.0% | 60.0% | 2.5 : 1 |
| ¾ pot | 30.0% | 42.9% | 57.1% | 2.3 : 1 |
| Pot | 33.3% | 50.0% | 50.0% | 2 : 1 |
| 1½× pot | 37.5% | 60.0% | 40.0% | 1.7 : 1 |
| 2× pot | 40.0% | 66.7% | 33.3% | 1.5 : 1 |
Pot of 100 before the bet. α = bet ÷ (pot + bet); minimum defence = 1 − α; value : bluff = (pot + bet) ÷ bet.
Look at the pot-size row: the bluff needs 50% and the defence needs 50%. Against a half-pot bet the defender continues two thirds of the time and the bettor's bluffs need to succeed one third. Solver folding frequencies on the river sit close to these figures because that is the point where neither side can gain by bluffing more or folding more.
MDF is a ceiling on folding, not a target
Before the river, hands have equity and there are more bets to come, so the pure formula no longer holds. Solvers routinely defend less than MDF on the flop when the bettor's range is stronger, and more when it is weaker. Use MDF as the reference point the solver is deviating from, and ask why.
The spot
Not sure? Count your outs in the draw odds explorer or compute it in the odds calculator.
What the math says
With 33% equity, calling loses in expectation.
Expected over many repetitions of this exact spot, assuming the call closes the action. Future streets, implied odds and reverse implied odds are not included.
Break-even equity by bet size
Facing a bet of this size, you need at least this much equity to call.
| Bet | Pot odds | Break-even | Bluff must work |
|---|---|---|---|
| ¼ pot | 5 : 1 | 16.7% | 20.0% |
| ⅓ pot | 4 : 1 | 20.0% | 25.0% |
| ½ pot | 3 : 1 | 25.0% | 33.3% |
| ⅔ pot | 2.5 : 1 | 28.6% | 40.0% |
| ¾ pot | 2.3 : 1 | 30.0% | 42.9% |
| Pot | 2 : 1 | 33.3% | 50.0% |
| 1½× pot | 1.7 : 1 | 37.5% | 60.0% |
| 2× pot | 1.5 : 1 | 40.0% | 66.7% |
“Bluff must work” is the same formula from the bettor’s side: how often a bet with no equity must win the pot outright to break even.
Quick check
An opponent bets $75 into a $100 pot. What is α, and how much of your range must continue to deny an automatic profit?
Show answer
α = 75 ÷ 175 = 42.9%, so MDF = 57.1%. Facing the bet, a call needs 75 ÷ 250 = 30% equity.