Poker odds by the numbers: hand probabilities, outs and pot odds

Poker hand probabilities

Hand rankings follow rarity: the scarcer a hand, the higher it sits. A five-card deal has 2,598,960 possible hands. Hold'em lets you pick your best five from seven cards, which gives 133,784,560 combinations and makes every strong hand far more common.

Two things stand out. In five cards, no pair is the most common result at 50.1%; with seven, you hold at least a pair 82.6% of the time, so a hand that has "nothing" by the river is below average. And a full house, a 1 in 694 shot in five cards, arrives about once in 39 seven-card hands. The poker odds calculator shows the formula behind every five-card count.

Starting hand odds

You get two cards from 52, so there are C(52,2) = 1,326 possible starting hands. Every figure below is the number of those hands that fit, divided by 1,326. Ace-king, for example, is 16 hands (four aces × four kings), four of them suited.

Pocket aces arrive once every 221 hands on average, so a player who sees 60 hands an hour waits well over three hours between them. Suited hands are less special than they feel: nearly a quarter of all starts are suited, and it adds only a few points of equity.

Flop odds

Once you hold two cards, 50 remain, and the flop is three of them: C(50,3) = 19,600 possible flops. The chance of flopping a given hand is the number of those flops that give it, divided by 19,600.

The set is the classic example. With a pair in your hand, two cards of your rank are left, and at least one lands on the flop 11.8% of the time, or 1 in 8.5. Unpaired cards pair at least one of themselves about a third of the time, so on roughly two flops in three neither of your cards has found a partner.

Outs and the odds of hitting

After the flop 47 cards are unseen, and an out is one that would make your hand. The chance of catching one on the turn is simply outs ÷ 47. By the river, work out the chance of missing twice, C(47 − outs, 2) ÷ C(47, 2), and take it from one.

A flush draw is 4.2 to 1 against on the turn but 35.0% with both cards to come. A pocket pair hoping to improve after missing the flop has only two outs, 8.4% by the river, which is why chasing it rarely pays. The rule of 2 and 4 gets close: 32% for an open-ended straight draw against an exact 31.5%.

Pot odds in one sum

Pot odds compare the cost of a call with the pot you would win. If the pot is £150.00 after an opponent's bet and it costs £50.00 to call, you are putting in £50.00 to win £200.00. The break-even point is £50.00 ÷ £200.00 = 25.0%. Win more often than that and calling makes money over many hands.

Put a flush draw against that £50.00 call. With one card to come it wins 19.1%, short of 25.0%, so calling a turn bet of that size loses on average. With two cards to come, if you are all in on the flop, 35.0% is well above it. Our Hold'em equity calculator does this sum for any two hands and any board, with the exact equity in place of an outs count.

Why poker has no house edge, only a rake

In roulette or blackjack the house is the other side of every bet, and the house edge is its margin on each one. In poker you play the other people at the table, the room holds no cards and no seat has a built-in advantage.

AI

The room earns its money from the rake: a percentage taken from each pot, usually with a cap, or a fixed entry fee in tournaments. As an example only, not any room's rate: at a 5% rake, a £200.00 pot pays £10.00 to the room and £190.00 to the winner. So a table of equally skilled players loses to the rake as a group, much as players lose to the house edge elsewhere. Rake rates differ between rooms, so check the stated rate.

Seven-card simulator

This simulator uses no money. It deals random seven-card hands, scores each one and counts how often every finishing hand appears, next to the exact share from the table at the top. The first 10,000 hands are shown already; add more and the gaps close.

With 1,000 hands, four of a kind may not appear at all; by 50,000 the common hands sit within about half a point of the exact figures. Results at the table swing the same way over a few hundred hands. If that swing has stopped feeling like a game, our responsible gambling page has free, confidential UK support.

Poker odds: questions people ask

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