Texas Hold'em Preflop: What Every Starting Hand Is Worth

You are dealt two cards. Before anything else happens, you have to decide whether to put money in. That decision is the most common one in poker and the easiest to get wrong, because the honest answer to “is this hand any good?” turns out to depend on things the question does not mention.

This page works it out from scratch. Every figure on it was computed for it — nothing is quoted from a book, and nothing is rounded off from memory — and every figure says whether it is exact or estimated, because those are different kinds of claim and mixing them is how people end up trusting numbers they should not.

1. The two kinds of number on this page #

Exact means enumeration: every possible case was visited and counted, one at a time. There is no error bar, because there is no sampling. When this page says a hand makes a flush on 6.383% of boards, that is 135,240 boards out of 2,118,760, counted.

Estimated means simulation: hands were dealt at random, millions of times, and the answer is the average. Every estimated figure carries its standard error — how far the estimate is typically off. With the sample sizes here that is about 0.08 percentage points, and the true value lies within about two of those, 95% of the time.

Where a figure could have been either, it is exact. Simulation appears only where exact counting is genuinely out of reach, and the page says so each time.

2. The game, in one section #

Each player gets two private cards. Then five community cards are turned face up in the middle, in three stages: the flop (three cards), the turn (one), and the river (one). Everything before the flop is preflop, which is what this page is about.

At the end, each player makes their best five-card hand out of the seven available to them — their two plus the five in the middle. They may use both of their own cards, one, or neither.

Hands are ranked by category, weakest to strongest:

Within a category, the higher cards win. Two pairs of kings beat two pairs of queens; if both players hold a pair of kings, the next card decides, and so on down. Only when all five cards match in rank is the pot split.

Notation. Ranks are 2 3 4 5 6 7 8 9 T J Q K A. A starting hand is written as its two ranks plus a letter: s for suited (both the same suit, like 9♣8♣ = 98s), o for offsuit (different suits, 9♣8♥ = 98o), and nothing at all for a pair (AA). The ace is both the highest card and, in the straight A-2-3-4-5, the lowest.

3. Why 169 hands and not 1,326 #

There are 52 cards and you get two of them, so the number of starting hands is the number of ways to choose 2 from 52:

C(52,2) = (52 × 51) / 2 = 1,326

But most of those are the same hand wearing different clothes. No suit beats another in Hold’em. Rename every club a heart and every heart a club, and you get a deal that is exactly as likely, in which everybody holds a hand of exactly the same strength against exactly the same opposition. So 9♣8♣ and 9♥8♥ cannot be told apart by anything that matters.

What survives that renaming is only two things: the two ranks, and whether the suits match. That leaves three families:

The 12 offsuit combinations: the high card can be any of 4 suits and the low card any of the other 3. The 6 pair combinations: choose 2 suits from 4, without order.

Those combination counts matter beyond the arithmetic. They are the weights to use whenever something is averaged over the 169 types, because a type that stands for 12 of the 1,326 hands should count twelve times as much as one that stands for 4. Several of the checks further down depend on exactly that, so the published data carries the count in a column rather than leaving the reader to look it up.

4. All 169 hands at once #

Here is the grid every poker player already knows how to read, with real numbers behind it. Rows and columns run from the ace down to the two. Pairs lie on the diagonal, same-suit hands above it, different-suit hands below.

Change what the shading means with the control above it. Hover, tap or arrow onto any cell for that hand’s figures, and open the table underneath to read all 169 as text.

A note on how that grid is drawn, since it is a chart and charts can lie. Colour carries one continuous quantity, so the scale is a single hue from light to dark — not a rainbow, which would invent categories the data does not have, and not red-to-green, which roughly one man in twelve cannot read. The cells carry each hand’s name and not its value: a number printed in all 169 cells is unreadable at that size. Every value is one hover away and all of them are in the table, because a colour scale on its own is not something everybody can read.

5. Counting the boards: 2,118,760 #

To work out what a hand can become, you count what the middle of the table can be. You hold two cards, so 50 remain, and five of them come out:

C(50,5) = 2,118,760

A reasonable objection: shouldn’t it be 50 × 49 × 48 × 47 × 46 = 254,251,200, the number of ways to deal five cards in order?

Both are correct counts of different things, and the ratio between them is exactly 5! = 120, the number of ways to shuffle five cards among themselves. The question is whether order matters for what we are counting, and for a finished hand it does not: a flop of 7♦ K♠ 2♣ followed by 5♥ and J♣ gives the same hand as a flop of 2♣ J♣ 5♥ followed by K♠ and 7♦. The 120 cancels between the favourable cases and the total, so the probability is the same either way and the unordered count is 120 times less work.

When order would matter: if you were analysing decisions street by street, because then it matters which cards are known when. For the finished hand at the river, it does not.

6. What a hand can become #

For each of the 169 hands, every one of those 2,118,760 boards was dealt and the resulting hand classified. These are counts, not estimates.

Take 98s, the suited connector:

The categories are exclusive — each board counts once, under the best hand it makes — so they sum to the total exactly. “Straight or better” is the bottom five added up: 17.462%.

6.1 The same number, by hand #

A program that counts 358 million things is easy to trust and hard to check. So here is the flush, worked out with a pencil, to see whether the program agrees.

You hold two clubs. Of the 50 cards left, 11 are clubs and 39 are not. You have two already, so you need at least three more clubs on the board. The cases do not overlap — a board has exactly three, or exactly four, or exactly five clubs — so they can simply be added:

Why multiply: each group of three clubs can be combined with any group of two non-clubs, and every such pairing is a different board.

And then the case almost everybody forgets. You also have a flush if the board brings five cards of some other suit — the flush is on the table and you play it. For each of the three other suits there are 13 cards and you take 5: 3 × C(13,5) = 3 × 1,287 = 3,861 boards. These cannot overlap with the cases above, because five diamonds leaves no room for three clubs.

135,597 + 3,861 = 139,458 boards
139,458 / 2,118,760 = 6.582%

Does the program agree? It counted 135,240 boards as “flush” and 4,218 as “straight flush” — and a straight flush is a flush too. 135,240 + 4,218 = 139,458. Exactly. (There is no flush hiding in the full house column, because seven cards cannot make both.)

6.2 “At least one” is easier backwards #

With 9-8, how often does the board bring a nine or an eight? There are three nines and three eights left, so six good cards and 44 others. Counting “at least one” means counting one, two, three… — but counting none is a single easy case: choose all five board cards from the 44 others.

P(none)        = C(44,5) / C(50,5) = 1,086,008 / 2,118,760 = 51.26%
P(at least one) = 1 − 51.26% = 48.74%

This is the complement rule, and it is worth internalising because “at least one” questions are everywhere in poker and the backwards version is almost always the easy one.

6.3 Why connectors make straights and aces do not #

There are ten possible straights: A-2-3-4-5, 2-3-4-5-6, … , T-J-Q-K-A. Your two cards help toward a straight only when both fit inside the same window of five consecutive ranks. Count the windows:

More windows, more ways to get there. A-K sits at the edge of the ladder and can only make the one straight at the top, which is why T9s reaches a straight or better far more often than AKs does — and why that fact, on its own, tells you nothing about which hand to play.

6.4 Not all pairs are alike #

A pocket pair always has the same chance of making a flush, since the suits do not care which rank they are. But straights are another matter: a ten or a five sits inside five of the ten possible straights, while an ace, a king or a two sits inside only two. So the middle pairs reach a straight more often:

That difference is real but small, and it is not the interesting one. The sharper version comes from asking what your two cards actually add to the board, which is section 10.

7. How we know the counts are right #

A program that miscounts one board in ten thousand shifts every number on this page a little and breaks nothing visibly. So the counting was tied to something computed independently, long before this code existed.

Take any set of seven cards. It can be split into “two private cards plus a five-card board” in C(7,2) = 21 ways, and the player’s category is the same in all 21, because it depends on the seven cards and not on which two were private. So adding up the counts over all 1,326 starting hands counts every possible seven-card hand exactly 21 times. Divide by 21 and you must land on the published number of seven-card hands of each category:

All nine match, with nothing left over when dividing by 21. A mistake in detecting straights, or flushes, or full houses would have to be cancelled by an exactly offsetting mistake somewhere else to survive that.

That check covers the categories. It says nothing about whether the program compares two hands of the same category correctly — whether it knows that a pair of nines with an ace beats a pair of nines with a king. That is checked separately, and exhaustively: there are exactly 7,462 distinct five-card hand values in poker, a number known for decades, and the evaluator was run over all 2,598,960 five-card hands to see how many different values it produced. It produced 7,462. An evaluator that ignored a kicker would merge two hands that should rank differently and come up short.

And finally, against somebody else’s code. Seven chosen matchups were counted twice over — once by this engine and once by treys, an independent evaluator — comparing wins, ties and losses as exact integers rather than comparing the equity, which could agree by luck while the three counts are wrong. All seven agreed on every count.

8. Potential is not the same as winning #

Everything above answers “what can this hand become”. It does not answer “should I play it”, and the gap between those two is where most poker intuition goes wrong.

32o reaches a straight or better 9.33% of the time. AKo reaches one 7.62% of the time. 32o makes more straights. It is also, by a distance, the worst hand in the deck. Making a category more often is worthless if the category you make is usually smaller than the one your opponent makes with the same board.

What matters is equity: the share of the pot a hand collects on average, if it is dealt out to the river and shown down. A showdown has three outcomes, and they are counted separately here because they are different facts:

win    your hand is strictly best                 you take 1
tie    k players hold equally best hands          you take 1/k
lose   somebody else is strictly better           you take 0

equity = (wins + sum of 1/k over the ties) / deals

The 1/k on a tie is not an approximation or a convenience. A tied pot really is split k ways, so 1/k really is what you collect. And ties are rarer than people think, because a better side card is a win, not a tie: the pot splits only when all five cards match in rank.

8.1 Every hand against every hand #

Equity against one opponent was computed exactly, for every ordered pair of the 169 hands — all 28,561 of them — by dealing out every possible board of every possible matchup. That is 161 billion hand evaluations, and it is the table everything else here is built on.

Notice 98s and 22, which are worth almost exactly the same — and are not the same hand at all. 98s ties twice as often. That is why this page reports wins, ties and losses separately everywhere: one equity figure hides it.

8.2 The check that constrains everything at once #

At a table where everybody holds a random hand, nobody has an advantage. The seats are symmetric, so all the equities must be equal, and since they are shares of one pot they must add up to 1. Each player is therefore worth exactly 1/(number of players).

So the average over all 1,326 starting hands, weighted by those combination counts from section 3, must come out at exactly one half. It does — and because every figure in the matrix is an integer count of boards rather than a decimal, that check is an equality between two whole numbers, with no rounding and no “close enough”. Both sides are 2,781,381,002,400.

That one check constrains all 28,561 cells together. A mistake anywhere in the evaluator, the counting or the weighting pushes some hands up and others down and the average off its mark.

9. The average hides the shape #

“AKs is worth 67% against a random hand” is an average, and behind it are 1,225 specific hands your opponent might hold. Against some of them AKs is a huge favourite; against others it is drawing nearly dead. The single figure says nothing about which.

So each hand was measured against every hand an opponent can hold, exactly. Define it carefully first, because “does this hand beat mine?” has no answer before the board comes out:

An opponent’s hand beats yours when your equity against that specific hand is below one half — that is, if the two were turned face up and dealt out every possible way, you would collect less than half the pot.

Nothing beats aces. Not one of the 1,225 hands an opponent can hold is a favourite against them; the only hand that does not lose is the other pair of aces, which splits. At the other end, 32o is a favourite against exactly two of them.

And look again at 98s, which has an equity of 50.80% — a hand that gets its fair share. It is behind two-thirds of the hands it can face. Its average is carried by the times it wins big, not by winning often. That is the whole argument for publishing a distribution instead of a mean.

10. What your cards actually add #

Here is a figure that sounds encouraging and is not: 72o makes two pair or better on 34% of boards. That sounds like a playable hand. It is the worst hand but one.

The catch is that most of those two pairs are sitting on the board, where every other player at the table has them too. A category you share with everybody is worth nothing. So: how often do your own two cards actually improve on what the board gives away for free?

The first way of asking turns out to be a dud, and it is worth seeing why. Require the best five cards to include at least one of your own, and 72o’s “two pair or better” falls from 34.24% to… 34.04%. Across all 169 hands the largest gap is 0.41 points. The requirement is nearly vacuous, because one of your cards riding along as a kicker satisfies it: the board’s two pair plus your seven as the fifth card does use your seven.

The question has to be sharper. Every board was sorted into one of four buckets, by comparing your best hand against the hand the board makes alone:

Aces and deuces improve the board’s category equally often — 94.72% of the time — and only one of them ever goes further. A deuce is never a bigger pair than the board’s pair and never a useful side card, so pocket deuces either make the category or contribute nothing whatsoever. Aces beat the board’s own combination on one board in twenty-two.

And 32o fails to improve on the board at all on nearly half of all boards, against 9% for 72o, because a three and a two lose to the board’s own side cards while a seven often plays as one.

11. More than one opponent #

Everything so far has been against one opponent. Against several, two things change, and only one of them is obvious.

The obvious one: your equity falls, because the pot is split among more hands that might improve. The less obvious one: the order changes.

Against one opponent, 32o is the worst hand in the deck and 72o is better than it. Against eight, 32o is worth more than 72o — 6.12% against 5.40%, a gap of fourteen standard errors, so it is not noise. Against a crowd you need a real hand to win, and 32o makes straights that 7-2 cannot (section 6.3), while 7-2’s marginally higher cards stop being worth anything.

These eight-opponent figures are estimated, not exact: counting every way to deal sixteen cards and a board is out of reach. Each is the average of 400,000 simulated showdowns, and each carries its own standard error: 0.075 percentage points for AA, 0.034 for 72o, and never more than 0.08 anywhere in the table. That is why the gap between 32o and 72o above can be called real rather than guessed at — it is fourteen times the uncertainty in it.

11.1 Your own two cards #

The grid works in hand types. A player holds two specific cards. Pick them here and see both at once — and watch what changing a suit does, which is nothing at all unless it changes whether the two match.

12. Is this hand worth playing? #

“Worth playing” needs a yardstick, and there is an obvious one. At a table of n players all holding random hands nobody has an advantage, so each is worth exactly 1/n of the pot. A hand worth more than that is pulling more than its seat.

Call that the fair share, and the ratio of a hand’s equity to it the fair-share index. Above 1.00 the hand is above average for that table; below it, below.

Two things fall out of that table that no fixed list of “good hands” can tell you. 22 is above average heads-up, below average six-handed, and above average again nine-handed — because at a crowded table the hands that beat it are busy beating each other, and a pair that holds up is worth more than high cards that do not improve. And 98s climbs steadily as the table fills, while 72o collapses.

You can shade the grid above by the fair-share index at any table size. It is a crude standard — it knows nothing about position, stack depth, or what the other players are doing — but it is a standard that can be computed, which is more than most advice offers.

12.1 Try it #

13. Playing against somebody who is not random #

Everything so far assumed your opponent holds any two cards. Real players fold their worst hands. So the figures above are the right answer to one question — what is this hand worth against an unknown hand — and the wrong answer to “what is this hand worth against that player”.

A range is the set of hands somebody is willing to play, and “top X%” is the shorthand: rank all 1,326 hands by strength and take the best X%. Ranking them needs an order, and the only order on this page that is computed rather than asserted is equity against a random hand, so that is the one used. The top 5% comes out as twelve hand types:

AA KK QQ JJ TT 99 88 AKs 77 AQs AJs AKo

Two things about that ranking are choices, not facts, and they are worth arguing with. It puts 88 and 77 inside the top 5% ahead of AQs and AJs, and it contains no suited connectors at all — because it knows nothing about position, stack depth, or what happens after the flop, all of which good players weigh. And ranges here are built from whole hand types, never split, so a range asked for 5% actually covers 5.43%: splitting a type would mean claiming somebody plays AKo from three suit combinations and folds the fourth, which nobody does.

What happens when the opponent gets choosy:

Aces get better as the opponent gets choosier — up to a point. AA is worth more against a top-20% range (85.68%) than against a random hand (85.20%), because a top-20% range is full of offsuit broadway cards that aces crush, while a random hand might be 65s, which has more ways to get there. Only against a very tight range does AA finally drop, and only to 82.49%.

Suited connectors fall apart. 98s goes from a fair-share hand against anything to 30.48% against a top-5% range. Against unknown hands it is average; against strong ones it is a dog.

And AKs loses a third of its value the moment the opponent is only playing premiums — from 67.04% to 47.61%, which is below a coin flip. Hold that thought.

14. The all-in, and a rule that does not survive it #

There is a piece of advice everybody repeats: against an all-in, call only with JJ+ and AK. It is worth checking, because checking it needs everything above plus one thing that is missing.

The missing thing is pot odds. Calling an all-in does not need 50% equity, because some of the money in the pot is not yours any more. Heads-up, blind against blind, with both players holding S big blinds:

You are the big blind. They are all-in for S.
Folding leaves you S − 1. Calling costs S − 1 more, for a pot of 2S.

  worth of a call  =  2·S·q − (S − 1)        q = your equity
  equity you need  =  (S − 1) / (2·S)

Never 50%, and the shorter the stacks the further below it, because the blinds are a bigger share of a smaller pot.

14.1 The verdict #

The rule is wrong in both directions, and which direction depends on the opponent and the stack.

Too loose against a tight shove. Against somebody moving in with only the top 5%, AKo is a profitable call only at 5 big blinds, and AKs only up to 20. Deeper than that, both lose money. The hands that beat AK against a tight range are precisely the hands a tight player shoves. The rule tells you to call and lose.

Too tight against anything else. Against a top-20% shove — still selective — 26 hands are profitable calls at 10 big blinds and 19 at 20, against the rule’s six. Every pair down to 66 or 77 is in there, along with most of the ace-broadways the rule throws away.

A fixed list of hands cannot be right, because it names neither of the two things that decide the answer: the opponent’s range sets your equity, and the stack depth sets the equity you need. What survives is the rule’s shape — pairs and ace-broadways really do dominate every calling range here, and against a tight shove nothing outside them calls profitably. It is a fair summary of which hands matter and a poor one of where the line falls.

14.2 One hand against another #

15. Estimated figures, and how far to trust them #

Most of this page is exact. The equity against two or more opponents is not, because counting every way to deal sixteen cards and a board is out of reach. Those figures come from simulation, and a simulation without an error bar is an opinion.

The error of an average falls as one over the square root of the number of trials:

standard error = √( variance / trials )

which is why quadrupling the work only halves the error. At 400,000 trials per figure the standard error here is at most 0.08 percentage points, and the true value lies within about two of those, 95% of the time.

That is the theory. The check is that it actually happens:

Quadrupling the trials cut the error by 1.82 and then 2.20 times, against the 2.00 predicted. And at every size, the error the runs actually have matches the error they report, within 10%. That was measured against the exact one-opponent figures from the matrix — the truth, not another simulation.

16. Two things that surprise people #

16.1 Your own cards change your opponent’s odds #

Cards you cannot see do not change anything: an opponent’s unseen hand is as unknown to you as the deck. But cards you can see do, including your own.

Holding AA, there is exactly one way left for an opponent to also hold aces — one combination out of the 1,225 available to them, instead of 6 out of 1,326. Your two aces removed five of the six. That is why the all-in tool above works out how often somebody folds from the actual count of hands they could hold given yours, rather than from a fixed percentage: holding AA they fold a top-5% range 95.5102% of the time, and holding 72o, 94.3673%.

This is also why an all-in preflop is not decided by the two cards. The hands are turned face up and the whole board is still dealt — flop, turn and river — and the best five of seven wins. The only way to win without a board is for everybody to fold.

16.2 Unequal stacks make more than one pot #

Every figure on this page assumes both players have the same stack. When they do not, a hand can create side pots, and the rule is that you can only win from each opponent as much as you yourself put in.

Three players go all-in: A has 20,000, B has 11,000, C has 20,000.

• Main pot: everybody can cover 11,000, so 3 × 11,000 = 33,000, and all three play for it.
• Side pot: A and C each have 9,000 left over, so 2 × 9,000 = 18,000, and only A and C can win it. B cannot: B never put in enough.

If B has the best hand, B wins 33,000 and the better of A and C takes the 18,000. Nothing on this page models that — the all-in tool assumes equal stacks — but the arithmetic above is all of it.

17. What this does not model #

Stating this plainly matters more than any figure above, because every number here is the right answer to a narrower question than “how should I play”.

• No betting. Hands here go to showdown. Real poker is mostly about getting opponents to fold, which nothing here measures.
• No position. Acting last is worth a great deal and is invisible here.
• No postflop play. A hand that plays well after the flop is worth more than its preflop equity; one that does not, less. That is most of the argument for suited connectors, and this page cannot make it.
• No unequal stacks, so no side pots (section 16.2), and no antes.
• No opponent who reacts. In the all-in tool the opponent’s range is a fixed input. That matters: at 10 big blinds against somebody calling the top 20%, every one of the 169 hands is a profitable shove, 32o included. That is the correct answer to the question asked, and it is not advice — a real opponent who noticed would call far wider and most of those shoves would stop working. Finding ranges that are stable against each other is a game-theory problem this page does not solve.
• No tournament prize structure. A chip is worth a chip here; in a tournament it is not.
• The ranking is a choice. Ordering hands by equity against a random hand is defensible and computed, but it is not how strong players rank hands (section 13).

18. Where the numbers came from, and how to check them #

Every figure on this page is produced by an engine built for it: holdem-preflop-equity. It rebuilds every table from nothing with one command, validates each one before writing it, and refuses to publish a table that failed its own check. Among those checks:

• The nine hand categories must reproduce the published seven-card hand counts, all nine, with no remainder (section 7).
• The evaluator must produce exactly 7,462 distinct five-card hand values.
• The head-to-head matrix must be its own mirror, and must average to exactly one half — checked as an equality between two integers.
• The distribution of “how many opponents beat you” is computed exactly for up to four opponents by inclusion-exclusion, and separately by brute force, walking every deal one at a time. The two agree.
• Seven matchups are counted again by an independent evaluator written by other people.

The data #

Everything this page draws is published, and documented field by field:

• hands.json — one record per starting hand: equity against 1 to 8 opponents and against every range, the nine potential categories, the four-way split of what your cards add, and the distribution against all 1,225 opponent hands.
• headsup-matrix.json — the exact 169 × 169 matrix, as wins and ties. All 28,561 cells.
• DATA_DICTIONARY.md — what every field means, and for each one whether it is exact or estimated.

Probabilities in those files are integers in ten-thousandths: divide by 10,000. The rounding is bounded by half a ten-thousandth, which is finer than anything shown here and finer than the simulation’s own error.

The fuller tables the engine writes — wide CSVs with the raw integer counts rather than rounded probabilities — are not committed, because they are generated and large. make rebuilds all of them in about twenty minutes.

If you think a number on this page is wrong, those files are where to start, and if you are right it is a bug worth reporting.