Probability of Poker Hands: How the Counting Works

The probability of poker hands is a matter of counting: how many five-card sets make a given hand, set against every five-card set the deck can deal. The fewer ways a hand can be built, the higher it ranks, which is why a royal flush, at 1 in 649,740 five-card deals, sits at the top of the order while one pair turns up about once in every 2.37 hands. The same order governs the seven cards a Texas Hold'em player draws from, though, as two mathematicians have shown, it does not hold forever as hands grow larger.

Where the numbers come from

Every figure in a probability table is a ratio of two counts. The bottom of the fraction never changes: it is the total number of distinct five-card hands a standard deck can produce. The top changes with each hand, and it is found by asking two plain questions. What does the hand require? In how many ways can the deck supply it?

The clearest worked set of answers sits in Neil E. Cotter's Conceptual Tools notes at the University of Utah, which take the five-card hands one by one. Read in sequence, they show that the ranking printed on every casino placard is not a matter of taste or tradition. It is arithmetic.

Royal flush

A royal flush is the ace, king, queen, jack and ten of one suit. Cotter's point is that once the ace is fixed, the other four cards are fixed with it, so only the suit is free to vary. A hand with a single free choice is about as scarce as a hand can be, and the 1 in 649,740 figure follows from that.

Four of a kind

Four of a kind is four cards of one rank plus any other card. The count takes two steps: choose the rank that supplies the four, then choose the odd card from whatever remains. The odd card has wide latitude, but the four matching cards have none, which keeps the hand near the top of the order.

Flush

The flush is where the counting earns its keep. Within a single suit, you count every way to choose five of its thirteen ranks. Some of those sets run in sequence, from ace-high down to five-high, and those are straight flushes, not flushes. Subtracting the ten straight-forming rank sets from each suit's total leaves the true flush count. A bare table of odds hides that step, yet it is the one that keeps the straight flush and the flush from being counted twice.

Three of a kind and one pair

Lower in the order, the counts swell quickly. Three of a kind arrives in about 1 in 47.3 five-card hands, and one pair in roughly 1 in 2.37. The difference between those two figures is the difference between a hand worth noticing and a hand a player sees most evenings at the table.

Poker probability hands chart

The chart reads from rarest to most common, and the rarest rows are the ones the deck can build in the fewest ways. Figures are for a five-card deal.

Between the flush and the rows below it sit the straight and the full house. Their places follow the same logic: in standard five-card frequency order, the straight is the most common of the three, then the flush, then the full house. That is why a full house beats a flush, and a flush beats a straight.

Do the probabilities change with seven cards?

The figures change, but the order does not, for any hand size a real game deals. Texas Hold'em builds each player's best hand from seven cards, and the question is whether the extra cards reshuffle the ranking. Wu and Wu took the problem to its limit in their paper on n-card poker, counting hands of every size.

Their answer is that a flush stays rarer than a straight for any hand of 11 cards or fewer and becomes more common only beyond that. The full house holds out longer still: it overtakes the straight only once a hand exceeds 19 cards. Seven cards sit comfortably inside both limits, so the Hold'em player can rely on the familiar order.

The finding also explains something that looks like a coincidence. The traditional ranking reads like a frequency table because, at five cards, it is one. Stretch the hand far enough and the two lists part company, which is a reminder that the hands of poker were ranked for the game as it is dealt, not for every game that could be.

From dealt hands to Hold'em tables

A frequency tells you how often a hand is dealt, not how often it wins. Whether a pair of queens holds up depends on how many opponents are drawing against it, and that is a harder count. In the ANZIAM Journal, Falletta and Woodcock approached it by simulation, modelling Hold'em games with two to 23 players and estimating each player's chance of holding the winning hand with partial and then fuller information.

Simulation is only as good as its agreement with known answers, and theirs held up: wherever exact analytic results exist, the estimates converged to three significant figures. The study chose Hold'em for a reason it states plainly, describing the game as the most commonly played variant in casinos and online. Its title, which sets Taylor Swift's grasp of the odds against James Bond's, is the only playful thing about it.

For the reader, the two kinds of figure belong together. The five-card chart says what the deck can produce. The table simulations say what that produces against real opposition. A royal flush at 1 in 649,740 and a three-handed pot are different questions, and the counting that answers the first is the foundation for the second.

From the reference desk

How rare is a royal flush?

In a five-card deal, about 1 in 649,740. Once the hand is defined as ace through ten of a single suit, the suit is the only thing left that can vary, so the deck can build it in almost no ways at all.

Is a flush harder to make than a straight?

Yes, in both five-card and seven-card play. The flush only becomes the more common of the two when a hand runs past 11 cards, which no standard game deals.

How often is one pair dealt?

About once in every 2.37 five-card hands, which makes it the ordinary furniture of the game rather than an event.

Can simulated Hold'em odds be trusted?

In the ANZIAM Journal study of Texas Hold'em, simulated estimates were checked against exact analytic results wherever those exist and agreed with them to three significant figures.

Bibliography

• Neil E. Cotter, University of Utah (Conceptual Tools). Probability Combinatorics Example 15: 5-card poker hands. Undated, accessed October 2026.
• Wu and Wu, Communications on Number Theory and Combinatorial Theory (arXiv). Big Two and n-card poker probabilities. 16 August 2023.
• Falletta and Woodcock, The ANZIAM Journal (Cambridge University Press). A simulation study of Texas hold 'em poker: what Taylor Swift understands and James Bond doesn't. 8 August 2018.