The Math That Runs Every Poker Decision
Expected value: the only score that matters
Every call, bet and fold has an expected value (EV): what it earns on average over many repeats. You call 50 to win a pot of 150 with a 30% chance of being good. EV = 0.3 × 150 − 0.7 × 50 = 45 − 35 = +10 chips. The call loses seven times in ten and is still right, because the three wins pay for the losses and more. Winning players pick the highest-EV option and let the cards do what they do.
Outs, and the rule of 2 and 4
An out is a card that turns your hand into the likely winner. A flush draw on the flop has nine outs (thirteen of a suit, minus the four you can see). To turn outs into a percentage without a calculator:
- On the flop, with two cards to come: outs × 4. Nine outs ≈ 36% (the true figure is 35%).
- On the turn, with one card to come: outs × 2. Nine outs ≈ 18% (true figure 19.6%).
The rule drifts high on big draws: fifteen outs is about 54%, not 60%. For ordinary draws it is close enough to use under a clock.
Pot odds: the price you are offered
Pot odds compare what you must call with what you win. The break-even share is call ÷ (pot after your call). The bet size sets it:
| Bet as a fraction of the pot | Equity you need to call |
|---|---|
| One third | 20% |
| One half | 25% |
| Two thirds | 29% |
| Full pot | 33% |
| Twice the pot | 40% |
Put the two tools together. Flush draw on the flop (35%) facing a pot-sized bet (needs 33%): a thin call. The same draw on the turn (about 20%) facing a half-pot bet (needs 25%): a fold, unless the next section rescues it.
Implied odds: the money that isn't in the pot yet
Sometimes the current pot does not pay for a call but the chips you will win when you hit do. Pocket fives flop a set about 11.8% of the time, roughly 7.5 to 1 against. Nobody offers you 7.5 to 1 on a preflop call, and when you do flop the set you don't always get paid, so the working rule is to call a raise with a small pair only when the effective stacks are around 15 times the call or more. Short stacks across the table mean small implied odds and a fold.
Equity: your share of the pot
Equity is the fraction of the pot that is yours on average if all the cards were dealt out now. Some matchups to know by heart:
- Aces against kings: about 82% to 18%.
- Any higher pair against a lower pair: about 80% to 20%.
- A pair against two overcards (queens against ace-king): about 57% to 43%.
- Ace-king against ace-queen: about 73% to 27%.
- Pocket aces against 7-2 offsuit, the best hand against the worst: only about 88%.
Preflop, poker never deals certainty. It deals prices, and the first list of numbers in this lesson tells you which prices to take.
Bluffing is arithmetic too
A bluff risks your bet to win the pot, so it breaks even when the opponent folds bet ÷ (pot + bet) of the time. A pot-sized bluff needs a fold half the time. A half-pot bluff, one time in three. A third-pot bluff, one time in four. The same table from the caller's side says a third-pot bet only has to be good 20% of the time to call. Both players are reading one bet from opposite ends.
Variance and the bankroll
Correct decisions lose all the time. A 35% draw misses twice in three; a 57% favourite loses four times in ten. Variance is the gap between what you deserved and what you got, and over thousands of hands it shrinks while skill does not. A bankroll exists so that variance can't take you out of the game: for cash games, keep at least 20 buy-ins for the stake you play, and step down when you drop below it. Ten buy-in downswings happen to good players. Judge yourself by the decision, not by the pot.
Where to practise it
The Odds Trainer and the Journey levels in this app drill each of these numbers against a clock, with play-money hands. Start with pot odds and the rule of 4: those two cover most of the calls you will ever face.