Poker hand rankings and probabilities

The ten poker hands from best to worst, with the exact number of ways to make each one. Type five to seven cards below to see which hand they make.

Best five-card hand

Full house, kings full of sevens

K♠K♥7♣7♠K♦2♥9♦ — rank 4 of 10 in the table below.

The hands, best to worst

Five-card counts are out of all 2,598,960 possible five-card hands. Seven-card counts are for the best five of seven cards, out of 133,784,560, which is what matters in Texas Hold'em.

#	Hand	Example	5-card ways	5-card chance	Odds against	7-card ways	7-card chance
1	Royal flush	A♠K♠Q♠J♠10♠	4	0.00015%	649,739 : 1	4,324	0.00323%
2	Straight flush	9♥8♥7♥6♥5♥	36	0.00139%	72,192 : 1	37,260	0.028%
3	Four of a kind	Q♣Q♦Q♥Q♠4♦	624	0.024%	4,164 : 1	224,848	0.168%
4	Full house	K♠K♥K♦7♣7♠	3,744	0.144%	693 : 1	3,473,184	2.60%
5	Flush	A♦J♦8♦6♦2♦	5,108	0.197%	508 : 1	4,047,644	3.03%
6	Straight	10♣9♦8♠7♥6♣	10,200	0.392%	254 : 1	6,180,020	4.62%
7	Three of a kind	8♠8♥8♦K♣3♠	54,912	2.11%	46.33 : 1	6,461,620	4.83%
8	Two pair	A♥A♣9♠9♦5♥	123,552	4.75%	20.04 : 1	31,433,400	23.50%
9	One pair	J♠J♦A♣7♥4♠	1,098,240	42.26%	1.37 : 1	58,627,800	43.82%
10	High card	A♣Q♦9♥6♠3♣	1,302,540	50.12%	1.00 : 1	23,294,460	17.41%

What each hand is

Royal flush: A-K-Q-J-10 all of one suit. The best possible hand.
Straight flush: five consecutive cards of one suit. The higher top card wins.
Four of a kind: four cards of one rank, plus any fifth card (the kicker).
Full house: three of one rank and two of another. The three decide first.
Flush: five cards of one suit, not consecutive. Compared card by card from the top.
Straight: five consecutive ranks in mixed suits. The ace plays high or low, never both.
Three of a kind: three of one rank and two unrelated cards.
Two pair: two different pairs and a kicker. Highest pair first, then the second, then the kicker.
One pair: two of one rank and three kickers.
High card: none of the above. Compared card by card from the top.

How the probabilities are counted

The five-card column is pure combinatorics. There are C(52, 5) = 2,598,960 ways to choose five cards from a deck. A few counts show how the rest are built:

Four of a kind: 13 choices of rank for the four, times 48 choices for the fifth card: 13 × 48 = 624.
Full house: 13 ranks for the three, C(4, 3) = 4 ways to pick their suits, 12 ranks for the pair and C(4, 2) = 6 ways to pick theirs: 13 × 4 × 12 × 6 = 3,744.
Straight flush: ten possible straights (five-high to ace-high) in four suits is 40, of which the 4 ace-high ones are royal flushes, leaving 36.
Flush: 4 × C(13, 5) = 5,148 single-suit hands, minus the 40 straight flushes: 5,108.

This site's test suite checks the whole five-card column by brute force: it deals every one of the 2,598,960 hands through the same evaluator that powers the checker above and the equity calculator, and requires the counts to match exactly.

The seven-card column counts every set of seven cards, C(52, 7) = 133,784,560 of them, by the best five-card hand they contain. These are the standard published figures; they are harder to derive by hand because a seven-card set can contain several candidate hands at once, such as a flush and a straight, and only the best one counts.

Worked example: the default cards

The checker opens on K♠ K♥ 7♣ 7♠ K♦ 2♥ 9♦, the kind of seven cards you might have at a Hold'em showdown with K♠ K♥ in your hand and the rest on the board. The evaluator sorts the ranks by how many times each appears: three kings, two sevens, and single cards. Three of one rank plus two of another is a full house, kings full of sevens, and the 2 and 9 play no part. If an opponent also had a full house, the three-of-a-kind part is compared first, so kings full of anything beats queens full of aces.

Where the rankings and odds do not apply

These are the standard high-hand rankings used in Texas Hold'em, Omaha, seven-card stud and five-card draw. Some games change them. In short-deck (six-plus) Hold'em a flush beats a full house, because removing the low cards makes flushes rarer. Lowball games such as razz and 2-7 triple draw rank the lowest hand best. Omaha uses the same rankings but requires exactly two hole cards and three board cards, which this checker does not enforce.

The probabilities also describe dealing, not play. They say how often seven random cards contain a hand, not how often a player shows one down at the river, which depends on who folded along the way.

Common questions

Does a flush beat a straight?

Yes. A flush beats a straight, and a full house beats both. The order follows how rare each hand is in five cards: there are 5,108 ways to make a plain flush and 10,200 ways to make a plain straight, so the flush ranks higher.

What happens when two players have the same hand?

The ranks decide it, then the kickers. Two pair of aces and nines beats aces and eights; if both players have aces and nines, the fifth card decides. Only the best five cards count, so a sixth or seventh card never breaks a tie. If the best five are identical in rank, the pot is split. Suits never rank in Texas Hold'em.

Is an ace high or low in a straight?

Both, but not at once. A-K-Q-J-10 is the highest straight and 5-4-3-2-A (the wheel) is the lowest, a five-high straight. A straight cannot wrap around the ace, so Q-K-A-2-3 is not a straight.

Why are the seven-card probabilities so different from the five-card ones?

With seven cards to choose five from, strong hands become far more common. A pair or better appears in about 82.6% of seven-card hands but only 49.9% of five-card hands. High card drops from about half of all five-card hands to about 17% of seven-card ones, so in Hold'em it is actually less common than one pair or two pair even though it ranks below both. The order is still the five-card one, by convention.

What are the odds of being dealt a royal flush?

In five cards, 4 in 2,598,960, or about 1 in 649,740. In Texas Hold'em, using the best five of your two cards and the five on the board, 4,324 in 133,784,560, which is about 1 in 30,940 hands that go to the river.