Poker Hand Probabilities, Derived Rather Than Memorised

Everything is a slice of 2,598,960

A five-card hand from a 52-card deck is an unordered selection, so the number of them is C(52,5) = (52 × 51 × 50 × 49 × 48) / (5 × 4 × 3 × 2 × 1) = 311,875,200 / 120 = 2,598,960. Every probability in poker is some subset of those hands divided by that total, and each subset can be counted by the same recipe: choose the ranks, choose the suits, multiply. Nothing beyond multiplication and the binomial coefficient is needed anywhere in this article.

Two of the counts are trivial and worth doing first, because they set the pattern. A straight flush is five cards in sequence in one suit: there are ten possible sequences, from A-2-3-4-5 up to 10-J-Q-K-A, and four suits, so 10 × 4 = 40. Four of a kind is all four cards of one rank plus any fifth card: 13 ranks × 48 remaining cards = 624. Notice that both counts are exact and complete on their own; nothing has to be subtracted, because a hand cannot be a straight flush in two ways or a four of a kind in two ways.

The seven that need a little care

A full house is a rank in triplicate and a different rank in duplicate: 13 choices for the triple, C(4,3) = 4 ways to pick its suits, 12 remaining ranks for the pair, C(4,2) = 6 ways to pick its suits. 13 × 4 × 12 × 6 = 3,744. The order matters here — three-of-rank-A with two-of-rank-B is a different hand from three-of-B with two-of-A, which is why the factor is 13 × 12 and not C(13,2). Three of a kind extends the same idea: 13 ranks × C(4,3) suits for the triple, then C(12,2) = 66 ways to choose two distinct kicker ranks and 4 × 4 suits for them, giving 13 × 4 × 66 × 16 = 54,912. The kickers must be distinct ranks, or the hand would be a full house.

Two pair uses C(13,2) = 78 for the pair ranks, because here the two pairs are interchangeable, times C(4,2)^2 = 36 for their suits, times 44 for the fifth card — 52 minus the 8 cards of the two chosen ranks. 78 × 36 × 44 = 123,552. One pair is 13 ranks × C(4,2) = 6 suit pairs × C(12,3) = 220 kicker ranks × 4^3 = 64 kicker suits = 1,098,240. High card is everything left: choose five distinct ranks that do not form a run, C(13,5) − 10 = 1,277, then any suit pattern that is not all the same, 4^5 − 4 = 1,020, giving 1,277 × 1,020 = 1,302,540. That last line quietly subtracts both exclusions at once.

The two exclusions, and the wheel

Four cards of one suit and a fifth of the same suit gives 4 × C(13,5) = 5,148 monochrome hands, but 40 of those are also straight flushes and would then be counted twice. Subtract them and the flush count is 5,108. The straight has the mirror-image problem: 10 rank runs, each card free to be any of 4 suits, gives 10 × 4^5 = 10,240, of which the same 40 are straight flushes, leaving 10,200. Skip either subtraction and your nine counts sum to 2,599,040 — eighty too many — which is exactly how you catch the error without knowing where it is.

The number ten deserves its own sentence. There are thirteen ranks, so a naive count of five-in-a-row gives nine sequences from 2-3-4-5-6 up to 10-J-Q-K-A. But the ace plays low as well as high, so A-2-3-4-5 — the wheel — is a tenth valid straight, while 2-3-4-5-6 counted from the top and Q-K-A-2-3 wrapping round the deck is not a straight at all. Get this wrong in the low direction and the straight count becomes 9,180 instead of 10,200; allow the wrap-around and it becomes 13,260. Both errors show up immediately in the sum, which is why the sum is worth computing.

The sum is the proof

40 + 624 + 3,744 + 5,108 + 10,200 + 54,912 + 123,552 + 1,098,240 + 1,302,540 = 2,598,960. Not approximately — exactly. That equality is the strongest check available, because the nine arguments were built independently of each other and of the total. If any one of them double-counted a hand, or missed a case, or applied the wrong binomial coefficient, the sum would miss. It does not miss, which means the nine categories partition all five-card hands with no overlap and no gap. Every published table that does not close on this identity has an error in it somewhere, and this is how to find out in ten seconds.

As a second, independent check, all 2,598,960 hands were enumerated in code and classified one by one. The program returned 40, 624, 3,744, 5,108, 10,200, 54,912, 123,552, 1,098,240 and 1,302,540 — the same nine numbers the algebra produced, and it also confirmed it had visited exactly C(52,5) hands. Two methods that share no assumptions and agree to the unit is as close to certainty as counting gets.

The ranking order is a consequence, not a convention

Read the count column downwards: 40, 624, 3,744, 5,108, 10,200, 54,912, 123,552, 1,098,240, 1,302,540. It is strictly increasing, and it is also the ranking order every poker rule book prints. That is not a coincidence and it is not an aesthetic choice by an early rule-maker. If a rarer hand lost to a commoner one, the game would reward the easier achievement, and any player who noticed could exploit it. The ranking is forced by the counting, which is why it is identical in every jurisdiction and every variant that uses five-card hands.

The relation is more than an ordering — it is a ratio. A straight flush is 15.6 times rarer than a four of a kind, which is 6 times rarer than a full house, which is only 1.36 times rarer than a flush. That last gap is the tightest in the table, which is why the flush-versus-full-house boundary is the one people most often misremember: 5,108 against 3,744 is a difference of 1,364 hands out of two and a half million. The pair-versus-high-card boundary at the bottom is almost a coin flip, 42.3% against 50.1%, and together those two categories cover 92.4% of all deals.

Seven cards is a different problem

Everything above describes a hand of exactly five cards. Hold'em gives each player seven and scores the best five, which is not the same calculation and does not have the same answers. The space grows to C(52,7) = 133,784,560, and every hand has to be evaluated by finding its best five-card subset rather than simply classified. Enumerating all of them gives 41,584 straight flushes, 224,848 four of a kinds, 3,473,184 full houses, 4,047,644 flushes, 6,180,020 straights, 6,461,620 three of a kinds, 31,433,400 two pairs, 58,627,800 one pairs and 23,294,460 high cards — summing, again exactly, to C(52,7).

Two of those numbers overturn the neat story above. A straight, at 6,180,020, is rarer than three of a kind, at 6,461,620 — yet it still beats it, because the ranking is fixed by the five-card counts, not the seven-card ones. And high card, which is the commonest five-card outcome at 50.1%, falls to 17.4% with seven cards, below two pair at 23.5%. Poker's ranking order therefore matches rarity exactly in five-card games and only approximately in seven-card ones. That is a real, documented inconsistency in the game, not an error in the arithmetic, and it is a good reason to be sceptical of any published seven-card table that has not been enumerated.

Frequently asked questions

Why does the flush count subtract 40?

Because the nine categories have to be mutually exclusive for their counts to sum to the total. Choosing five cards from one suit gives 4 × C(13,5) = 5,148 hands, and 40 of those five-card runs happen to be consecutive in rank, which makes them straight flushes. If you leave them in the flush bucket they are also in the straight flush bucket, and the nine counts add up to 2,599,040 instead of 2,598,960. Exactly the same 40 hands have to come out of the straight count, taking 10,240 down to 10,200. This is the single most common defect in published tables: one of the two subtractions gets applied and the other does not, and the error is eighty hands, invisible unless you add the column.

Is A-2-3-4-5 really a straight?

Yes, and it is the reason the rank runs number ten rather than nine. The ace is allowed to play as the lowest card as well as the highest, so the valid sequences run A-2-3-4-5, 2-3-4-5-6, and so on up to 10-J-Q-K-A: ten in total. This is standard in every major rule set. What is not allowed is wrapping around the top — K-A-2-3-4 and Q-K-A-2-3 are not straights, because the sequence has two ends and the ace can occupy either one but not bridge them. Drop the wheel and the straight count falls to 9,180; allow the wrap and it rises to 13,260. Neither figure lets the nine categories sum to 2,598,960, which is how you catch the mistake without arguing about the rule.

Why is a flush ranked above a straight?

Because it is rarer, and by a clear margin: 5,108 flushes against 10,200 straights, so a flush is almost exactly twice as hard to be dealt. The intuition that trips people up is that there are only four suits and thirteen ranks, so surely matching suits is easier — but a flush requires all five cards to agree on one of four things, while a straight only requires the five ranks to fall in a window of five out of thirteen and lets every suit be anything at all. The counting settles it: 4 × C(13,5) − 40 = 5,108 versus 10 × 4^5 − 40 = 10,200. Rankings in poker are never a matter of taste; every boundary in the table is a comparison of two counts.

Do these probabilities apply to Texas hold'em?

No. They describe a hand of exactly five cards dealt at random, which is five-card draw or stud, not hold'em. In hold'em each player forms the best five from seven cards, so the space is C(52,7) = 133,784,560 and every category becomes more likely — a full house goes from 0.14% to 2.60%, a flush from 0.20% to 3.03%. Enumerating all seven-card hands also produces two inversions worth knowing: a straight, at 6,180,020, becomes rarer than three of a kind at 6,461,620, and high card drops from the commonest outcome at 50.1% to 17.4%, below two pair. The ranking order does not follow, because it is fixed by the five-card counts. If you want hold'em numbers, use hold'em numbers; the five-card table is the wrong tool for that job.

How can I check a hand-probability table I found somewhere?

Add the hand counts. If they do not come to 2,598,960 exactly, the table is wrong, and the size of the discrepancy usually names the mistake. Off by exactly 40 means one of the two straight-flush subtractions is missing. Off by 1,020 means the wheel has been dropped from the straight count, because one rank run is worth 4^5 − 4 = 1,020 hands and 10,200 minus 9,180 is exactly that. Off by 3,060 in the other direction means the count has allowed sequences that wrap past the ace. If the table gives only percentages, multiply each by 2,598,960 and check the results are whole numbers; a genuine count always is. Percentages alone hide errors that integers cannot.