Introduction to Basic Poker Math

Most poker decisions come down to one question: does this call, bet or bluff make money over time? You do not need to be good at maths to answer it. You need four short formulas and a feel for what each one is actually telling you.

This guide covers equity, pot odds, alpha, minimum defense frequency (MDF), expected value (EV) and outs. Every number below is calculated in full, so you can check it yourself.

What is equity?

Every formula in this guide compares a price with one number: your equity. Equity is the share of the pot that belongs to your hand on average: how often it would win if all the remaining cards were dealt and nobody bet again. A tie counts as half a win.

The cleanest way to see it is to count. You hold 9♥︎8♥︎ on a K♥︎7♥︎2♣︎4♠︎ turn, and for this example we know your opponent has K♠︎Q♦︎. Eight cards are visible, so 44 can come on the river. Any of the 9 remaining hearts gives you a flush and the pot. The other 35 leave your opponent’s pair of kings ahead.

• Equity = 9 ÷ 44 = 20.5%.
• In a 20 bb pot, that is 20.5% × 20 = 4.1 bb of the pot on average. It is not what you win in any single hand: you either win all 20 bb or nothing.
• At the table you do not know your opponent’s exact cards, so real equity is measured against their range, the average over every hand they could have. That is what a solver or an equity calculator does for you.

Equity says how much of the pot your hand owns right now. Pot odds, below, say how much equity you need for a call to be worth its price.

Pot odds: how often your call has to win

Pot odds tell you the minimum share of the time your hand has to win for a call to break even. The formula:

The pot after your call includes everything: the pot before the bet, your opponent’s bet and your own call. Leaving the call out of the denominator is the most common mistake in poker maths, and it makes every call look worse than it is.

Worked example

You are in the big blind on the river. The pot is 6.5 bb and the button bets 3.2 bb (about half the pot). The app shows the pot as 9.7 bb, because the bet is already in it.

You risk 3.2 bb to win 9.7 bb. Required equity = 3.2 ÷ (9.7 + 3.2) = 3.2 ÷ 12.9 = 24.8%. If your hand wins more often than that, calling makes money. If it wins less, folding is better.

One thing this number does not tell you is how many hands you should call with. That is a different question, answered by MDF below.

Required equity for common bet sizes

The same formula for any bet size, with the bet written as a fraction of the pot:

Villain bets (% of pot)	Your call must win
10%	8%
20%	14%
33%	20%
50%	25%
66%	28%
75%	30%
100%	33%
125%	36%
150%	38%
200%	40%

Alpha: how often a bluff has to work

Alpha is the same idea from the bettor’s side. When you bluff with a hand that cannot win at showdown, alpha is how often your opponent has to fold for the bluff to break even.

In this spot you bet 6.5 bb into 6.5 bb, a pot-size bet. Alpha = 6.5 ÷ (6.5 + 6.5) = 50%. If your opponent folds more than half the time, the bluff makes money. If they fold less, it loses.

You bluff (% of pot)	Opponent must fold
10%	9%
20%	17%
33%	25%
50%	33%
66%	40%
75%	43%
100%	50%
125%	56%
150%	60%
200%	67%

Minimum defense frequency (MDF)

MDF flips alpha around. It is the share of your range you need to continue with, by calling or raising, so that a bluff with no equity does not automatically make money against you.

Facing the pot-size bet above, MDF = 1 − 50% = 50%. In the pot odds example (3.2 bb into 6.5 bb), alpha is 3.2 ÷ 9.7 = 33.0%, so MDF is 67.0%. Notice that this is a very different number from the 24.8% required equity for the same bet.

The difference is hand versus range. Required equity is about one hand: can this specific hand call? MDF is about your whole range: how many of your hands should keep going? Both come from the same bet, and it is worth seeing how they move as the bet gets bigger:

Where MDF stops being the answer

• It assumes your opponent’s bluffs have no equity. That is close to true on the river. On the flop and turn their bluffs still have outs, so the simple formula does not hold.
• It ignores raises and what happens on later streets.
• It is a reference point, not a solver strategy. Solver outputs often defend less or more than MDF depending on ranges, board and position, so treat MDF as a starting point.
• If you know your opponent bluffs far less than balanced, folding more than MDF suggests is an exploitative adjustment, not a GTO one. Our article GTO poker vs. exploitative poker explains the difference.

Expected value (EV)

EV is the average result of a decision if you made it many times. For a call:

Back to the river call: reward 9.7 bb, risk 3.2 bb. At exactly 25% equity, EV = 0.25 × 9.7 − 0.75 × 3.2 = 2.425 − 2.4 = +0.025 bb. That is almost exactly zero, which makes sense: 25% is right next to the 24.8% break-even point.

• A hand that wins 15% of the time: 0.15 × 9.7 − 0.85 × 3.2 = −1.27 bb. Fold it.
• A hand that wins 40% of the time: 0.40 × 9.7 − 0.60 × 3.2 = +1.96 bb. Call it.

This formula uses win% directly, which works on the river where nothing is left to come. On earlier streets your hand does not always get to showdown, so it realises only part of its equity. Our guide to equity realisation (EQR) covers that step.

Outs: your odds of improving

With a draw, the question is how often you will hit. An out is any unseen card that improves you to what is likely the best hand. A flush draw has 9 outs: 13 cards of the suit minus the 4 you can see.

On the flop you can see 5 cards (your 2 and the board’s 3), so 47 are unseen, since your opponent’s hand is not known. Your chance to hit on the turn, and to hit by the river if you see both cards:

Draw	Outs	Turn only	By the river
Gutshot	4	8.5%	16.5%
Open-ended straight draw	8	17.0%	31.5%
Flush draw	9	19.1%	35.0%
Flush draw + gutshot	12	25.5%	45.0%
Flush draw + open-ended draw	15	31.9%	54.1%

At the table there is a shortcut, the rule of 2 and 4: multiply your outs by 2 for one card to come, or by 4 for two cards. It is close with few outs and drifts with many:

Putting outs and pot odds together

You hold a flush draw on the flop and your opponent bets the size of the pot. Required equity is 33.3%. By the river you would hit 35.0% of the time, which looks like enough. But that 35.0% assumes you see both cards for this one price. You only see the turn for this call, and that hits 19.1% of the time.

If there is more betting to come, you are not getting 35.0% for your money. The call only works if you win enough extra on the times you hit, which is what implied odds are about. If you are all-in, there is nothing left to bet, and 35.0% is the right number.

Quick recap

• Equity: how often your hand wins the pot. Count the cards that win, divide by the cards that can come.
• Pot odds: call ÷ (pot after your call). Half-pot bet → your call needs 25%.
• Alpha: bet ÷ (pot + bet). Half-pot bluff → needs 33% folds.
• MDF: 1 − alpha. Half-pot bet → defend about 67% of your range. A reference point, not a solver strategy.
• EV: win% × reward − lose% × risk. Above the break-even equity, calls make money.
• Outs: outs ÷ 47 for the turn, 1 − (47 − outs)/47 × (46 − outs)/46 for both cards. Rule of 2 and 4 at the table.

New to counting combos? Start with Poker starting hands: 1,326 combos, 169 hands. When you want to check a real spot, the solver gives you the exact answer for every hand in your range.