Starting hands: how often and how far ahead

Two hands are face up before the flop, before any of the five shared cards is dealt: A♠ A♥ against K♠ K♥. What share of the pot do the aces take, on average over every board?

The long part of the bar is the aces’ share of the pot over every board. The grey line under it is one run of boards dealt at random: a sample, which lands near the exact share and shifts a little from one run to the next.

Leoformer dealer · rules and maths

I dealt hold’em for years, and the deal players remembered was the one where the aces did not hold up. The aces were still the hand ahead when the chips went in. Being ahead before the flop is a share of the pot, counted over every board the deck can make.

How often a starting hand is dealt

In Texas hold’em each player is dealt two cards face down, the starting hand, and five shared cards come later for everyone. Two cards from a 52-card deck can be dealt 1,326 ways, and when the order of the two cards and which suit is which stop mattering, those ways fall into 169 starting hands.

• Any pair1 in 17Two cards of the same rank, any rank.
• A pair of aces1 in 221One exact pair; every other exact pair comes as often.
• Ace and king1 in 82.9An ace and a king, of one suit or of two.
• Both cards of one suit23.5%Of all starting hands, any two ranks.

Each figure is a count of the ways two cards can be dealt. Any pair comes once in 17 deals on average; one exact pair, such as aces, is 13 times rarer.

All the starting hands on one grid

The grid below holds every starting hand once, in 13 rows by 13 columns. Pairs run down the diagonal, hands with both cards of one suit sit above it, and hands with two different suits sit below it. T stands for a ten.

• AAA pair, on the diagonal: 13 hands, each made 6 ways, each dealt 1 in 221.
• AKss for suited, both cards of one suit, above the diagonal: 78 hands, each made 4 ways, each dealt 1 in 331.5.
• AKoo for offsuit, two different suits, below the diagonal: 78 hands, each made 12 ways, each dealt 1 in 110.5.

The grid says how often each starting hand is dealt, and nothing about whether to play it.

Which hands to play

This course gives no advice of its own on which starting hands to play, and none of the documents this page quotes is a guide to it: the papers the counts come from count hands, and the Argentine rulebooks for the poker room say how a hand is dealt and settled. The grid above says how often each starting hand comes, and the next section says how far ahead one hand is of another.

Which hands to keep and which to fold is the player’s own choice; the figures on this page say what each choice starts from, not which one to make.

How far ahead a hand is before the flop

The share of the pot a hand holds before the flop, called its equity, is counted over every possible board, the five shared cards: on a board where the hand ranks higher, the pot counts in full, and on a board where the two hands are equal, the hand’s part of the split pot counts. With two hands face up, 48 cards are left, and every five-card board they can make is counted.

None of these four is lopsided the way aces against kings are: the first hand holds from 40.9% to 52.6% of the pot.

Suits move the figure a little. 2♣ 2♦ against A♠ K♥, the same ranks as the row in the table with no suit shared between the two hands, holds 53.0%, because which cards are already out changes which flushes each hand can still make.

A shared card and a higher kicker

When two starting hands share one card, the other card in each hand, the kicker, decides most boards: if the shared card pairs on the board, both hands pair it, and the higher kicker ranks above the lower one. How a kicker settles the pot at the showdown is lesson 7.5.

The queen is in both hands. King-queen moves ahead mainly when the board pairs the king and not the ace, and that leaves king-queen about a quarter of the pot.

A pocket pair and the flop

A pocket pair is a starting hand of two cards of the same rank, and two more cards of that rank are left in the deck. The flop, the first three shared cards, brings at least one of them in about one deal in 8.5, which gives the hand three or four cards of its rank.

• At least one more card of your pair’s rank on the flop11.8%Of all flops, for any pocket pair.
• Exactly one more card of your rank11.5%The same flops, without the ones that bring both.

The gap between the two figures is four of a kind on the flop, when both remaining cards of the rank come.

A hand far ahead before the flop still gives up the pot part of the time, and a stronger starting hand only gives it up less often. Knowing these figures is part of knowing the game, and no amount of knowing them makes any one deal come out a certain way.

Mayaformer floor supervisor · manners, money and breaks

The rare hands are rare by arithmetic, and waiting for them is how an evening at the poker table stretches. Waiting for good hands is patience; waiting past the time you set for the evening is not.

Pick the time you will stand up before you sit down, and keep to it whether or not the hands you were waiting for have come.

The lesson on one card

That is the lesson. Here it is on one card: print it, or keep it on your phone.

Practice: who is ahead before the flop?

Every deal shows two starting hands face up, before any shared card. You choose the range your hand’s share of the pot falls into. Level 1 deals a pair against a pair, level 2 a pair against two unpaired cards, and level 3 two hands that share one card.

Nothing is bet here, and no board is dealt into the trainer: each deal stops before the flop, and the answer comes with the exact share.

Check yourself

Mayaformer floor supervisor · manners, money and breaks

Three quick ones. Pick an answer, then check it. No timer, no points: just whether it stuck.

Back to the question we opened with. The aces take 82.6% of the pot against the kings before the flop, and the kings hold the rest.

Further reading

18+ under each province’s rules · Play responsibly · the helplines are further up this page

Further reading

• Counting starting poker hands · Brian Alspach, Simon Fraser UniversityHow the 1,326 two-card hands fall into 169 distinct starting hands.
• Cracking Aces: Part II · Brian Alspach, Poker Digest (2000)A mathematician’s count of how often a pair of aces holds up against nine opponents.
• Algorithms and assessment in computer poker · Darse Billings (PhD thesis, 2006)The 1,326 starting hands, and why many thousands of games are needed before a result can be trusted.

How it’s worked out