Poker Math Basics: Odds, Outs, Probabilities and EV
Poker maths is a small set of exact numbers and three formulas that tell you how often things happen and whether a call is profitable. You do not need advanced mathematics: counting hand combinations, converting outs to percentages and comparing a price with your chance cover most decisions. Every figure on this page was calculated exactly by listing every possible deal (all 133,784,560 seven-card hands and all 1,712,304 boards for each matchup), not estimated.
The short version
- There are 1,326 possible starting hands: 78 pairs (5.9%), 312 suited hands (23.5%) and 936 offsuit hands (70.6%).
- Outs to percentage: on the flop multiply outs by 4 for two cards to come (slightly too high above 8 outs), by 2 for one card.
- Pot odds: the share of the final pot you must put in is call ÷ (pot + call). Call when your chance of winning is higher.
- Expected value (EV) is the average result of a decision over many repetitions: win chance × win − lose chance × lose.
- Even the best starting hand loses often: aces win 81% against kings, and a hand like king-ten is far from "safe".
Counting starting hands
A deck has 52 cards and you hold two, so there are 52 × 51 ÷ 2 = 1,326 different starting hands. They fall into three shapes:
| Shape | Combinations per hand | Examples |
|---|---|---|
| Pair | 6 | A-A, 7-7 |
| Suited | 4 | A♠K♠, 7♥6♥ |
| Offsuit | 12 | A♠K♥, 9♦2♣ |
From this, the chance of being dealt a particular hand:
- Any pocket pair: 78 ÷ 1,326 = 5.88% (1 in 17).
- A specific pair such as aces: 6 ÷ 1,326 = 0.45% (1 in 221).
- Ace-king suited: 4 ÷ 1,326 = 0.30%; offsuit 12 ÷ 1,326 = 0.91%; either way 1.21% (1 in 83).
- Any two suited cards: 23.5%; any two offsuit cards: 70.6%.
These counts are also the basis of ranges: when you put an opponent on "queens or better", that is 18 combinations (6 each of Q-Q, K-K, A-A), and ace-king is 16. See range charts and starting hands.
What the flop does
With 50 unseen cards, the flop is one of 19,600 equally likely three-card sets. Exact chances for common situations:
- You hold a pocket pair and flop a set (at least one of the two remaining cards of your rank): 11.8%, about 1 in 8.5.
- You hold two unpaired cards and the flop contains at least one card of either rank: 32.4%.
- You hold two suited cards and flop a flush: 0.84%; exactly four to a flush (a flush draw): 10.9%.
So most flops miss most hands. That is why continuing with weak hands costs money over time.
Final hand frequencies
In Hold'em the best five cards out of seven count. Listing all 133,784,560 seven-card hands gives:
| Best hand | Combinations | Chance |
|---|---|---|
| High card | 23,294,460 | 17.41% |
| Pair | 58,627,800 | 43.82% |
| Two pair | 31,433,400 | 23.50% |
| Three of a kind | 6,461,620 | 4.83% |
| Straight | 6,180,020 | 4.62% |
| Flush | 4,047,644 | 3.03% |
| Full house | 3,473,184 | 2.60% |
| Four of a kind | 224,848 | 0.168% |
| Straight flush (incl. royal) | 41,584 | 0.031% |
These are not the five-card percentages in most charts: with seven cards to choose from, flushes and full houses are noticeably more common, and the chance of finishing with only a high card is 17% instead of 50%. The order of hands is explained in hand rankings.
Preflop matchups: how far ahead is "ahead"
Equity is your share of the pot if everyone's cards were shown and the board dealt out. Exact values over all 1,712,304 boards (suits chosen so that the hands share none, since shared suits change the result slightly):
| Matchup | First hand wins (incl. ties' share) |
|---|---|
| A-A vs K-K | 81.3% |
| A-A vs 7-2 offsuit | 87.4% |
| K-K vs A-Q offsuit | 71.6% |
| A-K offsuit vs A-Q offsuit (dominated) | 74.7% |
| 9-9 vs A-K offsuit (a "coin flip") | 55.7% |
| A-K offsuit vs 10-9 suited | 58.8% |
| A-K suited vs Q-Q | 46.2% |
| A-K offsuit vs Q-Q | 43.2% |
| 7-6 suited vs A-K offsuit | 42.3% |
| J-10 suited vs A-K offsuit | 41.2% |
Three lessons. A pair against two overcards is only a slight favourite (the classic coin flip is 55:45, not 50:50). Domination is brutal: ace-king beats ace-queen three times out of four, because the queen hardly ever helps. And even the best hand loses almost one time in five, so a single result says little about a decision; see variance in poker. To test any matchup yourself, use the poker odds calculator.
Outs and percentages
An out is a card that improves your hand to the best one. Counting outs on the flop (47 unseen cards) gives the exact chance of hitting by the river, and the rule of 2 and 4 is a shortcut:
| Outs | Typical draw | Next card (turn) | Turn or river | Rule of 4 |
|---|---|---|---|---|
| 2 | Pair to set | 4.3% | 8.4% | 8% |
| 4 | Gutshot straight | 8.5% | 16.5% | 16% |
| 6 | Two overcards | 12.8% | 24.1% | 24% |
| 8 | Open-ended straight | 17.0% | 31.5% | 32% |
| 9 | Flush draw | 19.1% | 35.0% | 36% |
| 12 | Flush draw + gutshot | 25.5% | 45.0% | 48% |
| 15 | Flush draw + open-ended | 31.9% | 54.1% | 60% |
The rule of 4 is accurate up to about 8 outs and overstates large draws. Above 8 outs, subtract (outs − 8) from the result: 15 outs → 60 − 7 = 53%. More in outs and equity.
Pot odds and expected value
Pot odds are the price the pot offers. If the pot is 80 and an opponent bets 20, the pot is 100 and you must call 20 to win 120 in total. Your call is 20 ÷ 120 = 16.7% of the final pot, so you need to win more than one time in six.
The formula: required equity = call ÷ (pot after the bet + call). Compare it with your chance from the table above. A flush draw (35% by the river, 19% for one card) clears 16.7% easily, a gutshot on one card (8.5%) does not.
Expected value puts it in one number: EV = (win chance × amount won) − (lose chance × amount lost). Calling 20 with a 35% chance to win 100 is 0.35 × 100 − 0.65 × 20 = +22 on average. Not every call wins, but over many repetitions positive EV decisions make money and negative ones lose it. Try your own numbers in the pot odds calculator, and see pot odds and implied odds for the extra money you can win later in the hand.
Maths that wins money without a calculator
- Fold equity. A bet wins in two ways: the opponent folds, or you win at showdown. A bluff only needs to work bet ÷ (pot + bet) of the time: a pot-sized bet needs 50%. See fold equity and the fold equity calculator.
- Bankroll. Even a winning player has losing months, so the number of buy-ins you hold must match your variance: bankroll management and the bankroll calculator.
- Tournament money is not linear. Chips lose value as you accumulate them, so tournament decisions near the money use ICM and the ICM calculator.
Frequently asked questions
What are the odds of being dealt pocket aces?
6 combinations out of 1,326, or 0.45%, which is 1 in 221 hands.
What is the chance of flopping a set with a pocket pair?
11.8%, about 1 in 8.5 times.
How do I convert outs to a percentage?
On the flop multiply the outs by 4 for the chance to hit by the river (by 2 for one card). Above 8 outs subtract (outs − 8). The table above has the exact values.
How often does a pocket pair beat two overcards?
About 55.7% for nines against ace-king offsuit, so it is a slight favourite, not 50:50.
What are pot odds in poker?
The call amount divided by the total pot after your call. It is the minimum chance of winning that makes the call break even.
What to read next
- Pot odds: the price of a call in detail.
- Outs and equity: counting draws and discounting outs.
- Implied odds: money you can still win.
- Starting hands: which of the 1,326 to play.
- Variance in poker: why the maths needs thousands of hands to show.
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