Pot odds, equity and EV: the arithmetic behind every call

This is the least glamorous page on the site and the one that pays for itself fastest. Everything here is arithmetic you can do in your head at the table, and every number below is worked out step by step so you can check it.

Pot odds: what a call needs to win

Pot odds are the price you are being offered. The formula is short:

required equity = amount to call ÷ (pot + opponent's bet + your call)

An example, and it is worth following the numbers rather than the words. The pot is 10bb. Your opponent bets 5bb. The pot is now 15bb, and it costs you 5bb to call — so you are calling 5 to win 20:

1. Amount to call: 5bb.
2. Pot after your call: 10bb (original) + 5bb (their bet) + 5bb (your call) = 20bb.
3. Required equity: 5 ÷ 20 = 0.25, so 25%.

In other words: if your hand wins more than a quarter of the time, calling makes money. If it wins less, folding is better — and the hand you fold is irrelevant, because this calculation never asks what you hold, only what it is worth.

Equity, and counting outs

Equity is how often your hand wins at showdown from here. When you are drawing, you estimate it by counting outs — the cards that make your hand — and the shortcut everyone uses is the rule of 2 and 4: multiply your outs by 4 with two cards to come (the flop), and by 2 with one card to come (the turn).

A flush draw has 9 outs. On the flop the rule gives 9 × 4 = 36%, and the real number is 34.97% — close enough to act on. On the turn the rule gives 9 × 2 = 18% against a real 19.57%: this time it is slightly low, which is the honest summary of the shortcut. It is an approximation in both directions and it drifts as the number of outs grows, so treat anything past nine or ten outs as needing a discount.

You can also arrive at the same answer by counting the cards that miss you, which is where the real numbers above come from: with 9 outs on the flop there are 38 cards that do not help from 47 unseen, and the chance of missing twice in a row is (38/47) × (37/46) = 65.03% — so you make it 34.97% of the time.

EV: the number that settles it

Equity tells you how often you win. EV tells you whether calling is worth it, which is a different question. The formula:

EV = equity × (pot + opponent's bet) − (1 − equity) × amount to call

Same spot as before — pot 10bb, they bet 5bb, so you call 5 to win 20:

• With 30% equity: EV = 0.30 × 20 − 0.70 × 5 = 6.00 − 3.50 = +2.50bb.
• With 25% equity: EV = 0.25 × 20 − 0.75 × 5 = 5.00 − 3.75 = +1.25bb.
• With 20% equity: EV = 0.20 × 20 − 0.80 × 5 = 4.00 − 4.00 = 0.00bb.

Look at the last line: 20% equity is exactly break-even, and it is exactly the pot-odds number from the first section. That is not a coincidence — the two formulas are the same statement, and the point of the second one is that it gives you a number in big blinds instead of a percentage, which is what a session review can add up.

Why EV beats results

A positive-EV call loses money on this particular hand fairly often — that is what 30% equity means. A negative-EV call wins sometimes, and the times it wins are the ones people remember. If you judge decisions by the result of the hand, you will learn to repeat the mistakes that happened to work, and to avoid the correct plays that happened to lose.

This is exactly why the trainer grades decisions against the equilibrium rather than against the outcome. A grade says what your decision was worth at the moment you made it, and it is the only kind of feedback that compounds.

Implied odds and reverse implied odds

Pot odds assume the hand ends now, and it does not:

• Implied odds — money you expect to win on later streets when you hit. Calling with a draw can be correct even when the immediate price is not, because a made hand will often get paid again.
• Reverse implied odds — the mirror image, and the one that costs people money. You hit your draw and still lose: the small flush against a bigger flush, the straight that completes a better straight, the pair that improves your opponent's two pair. Here you get paid nothing extra when you are ahead and lose another bet when you are behind.

The practical consequence: draws to the nuts can be called thinner than draws to second-best hands, and a hand that will be hard to play after it improves is worth less than its outs suggest.

Using this in the trainer

The trainer shows win probability and EV for the decisions you face, which makes it a place to test your estimates rather than just a place to be told the answer. Before you act, guess the number: is this call getting the right price? Then look at the panel and see how far off you were.

Missing by a little is normal. Missing by a lot means the input was wrong — often you counted too many outs (an opponent's range takes cards away from you), or you ignored that the hand continues on the next street. That difference is worth more than the grade itself, because the estimate is the thing you take to a table where there is no panel. The bluffing side of the same arithmetic is in bluffing frequencies and MDF, and the panel mechanics are in how to use this trainer.