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Poker maths

Expected value (EV)

The idea behind every good poker decision - what a play earns on average - with examples for calls, bluffs and all-ins.

Expected value, or EV, is what a decision wins or loses on average if you could make it thousands of times. Poker is a game of making positive-EV decisions and letting the results take care of themselves. You will lose plenty of hands you played well, and that is fine.

The formula

EV = (chance you win × what you win) - (chance you lose × what you lose)

A call

The pot is 150 after your opponent bets 50. You think you win 30% of the time.

  • You win 150 (the pot, including their bet) 30% of the time: +45.
  • You lose your 50 call 70% of the time: -35.
  • EV = +10. The call makes 10 chips on average, even though you lose more often than you win.

This is exactly what pot odds told you in shortcut form: you needed 25%, you have 30%.

Bluffs and fold equity

A bluff makes money when your opponent folds often enough. If you bet B into a pot of P, you risk B to win P, so the bluff breaks even when your opponent folds:

B ÷ (P + B) of the time.

How often a pure bluff must work to break even
A third of the pot 24.8%
Half the pot 33.3%
Two thirds 39.8%
Three quarters 42.9%
The pot 50.0%
Twice the pot 66.7%

Small bluffs need to work less often, which is why they are popular on boards your opponent probably missed. A semi-bluff - a bet with a draw - does even better: when it is called, you can still hit.

All in for the tournament life? Same maths

Your opponent shoves 1,000 into a pot that already holds 600. You must call 1,000 to win 1,600 + 1,000 = 2,600. You need 1,000 ÷ 2,600 = 38.5%. If your hand is a 45% coin flip, the call is +EV, whether you win this one or not.

Results are not decisions

Getting A♠A♥ all in against K♠K♦ and losing is a good decision with a bad result. Calling off your stack with 7♦2♣ and winning is a bad decision with a good result. Over thousands of hands the luck evens out and the decisions remain. Judge your play by the EV of your choices, not by the last pot.

Variance

Even a solid winning player has long losing stretches. A strategy that wins a little over many hands can easily lose for an evening, a week or longer. That is variance, and handling it is part of the game - see bankroll management and tilt.