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Poker maths

Preflop matchups

How often a pair beats two overcards, why ace-king is a coin flip, and the matchups worth knowing before you go all in.

When the chips go in before the flop, the result is down to the five board cards, and the odds are fixed. A handful of standard matchups cover almost every all-in you will see. Learn their shapes rather than the exact numbers.

The matchups that matter

All in before the flop: how often each hand wins approximate; the gap in a bar is ties
A♠A♥ 82%18% K♠K♦
K♠K♦ 81%18% Q♠Q♥
J♥J♦ 57%43% A♠K♣
Q♠Q♦ 54%46% A♥K♥
2♠2♦ 52%47% A♥K♣
9♠9♣ 70%30% A♥10♦
A♥K♦ 74%22% A♣Q♠
A♥K♦ 61%39% J♣10♠

The shapes behind the numbers

  • Higher pair vs lower pair: about 80/20. The underdog needs to hit one of two cards to make a set. AA vs KK and KK vs QQ are the classic coolers.
  • Pair vs two higher cards: about 55/45, a "race" or "coin flip". The pair is a small favourite, whether it is J♥J♦ or 2♠2♦, against something like AK. This is the most common all-in in poker.
  • Pair vs one higher and one lower card: about 70/30. 9♠9♣ against A♥10♦ only fears the aces.
  • Dominated hand: about 70-75/25. A♥K♦ against A♣Q♠ shares the ace, and AQ can only win by pairing the queen. A lot of ties too, when neither improves.
  • Two higher cards vs two lower cards: about 60/40. A♥K♦ is a favourite over J♣10♠, but not by as much as people think: connected cards have straight chances.

Why AK is not a big favourite

Ace-king is the best unpaired hand, but it is still just ace-high until it pairs, and it pairs on the flop only about a third of the time. Against any pair, even deuces, it is a slight underdog. That is why "AK or a pair?" arguments go on forever: whoever has the pair is ahead before the flop, and whoever has AK has a hand that plays better after it.

Using it at the table

When you decide whether to call an all-in, ask: what would I be up against? If you hold Q♠Q♦ and a tight player moves all in, their range is mostly AA, KK and AK. You are an 80/20 dog against two of those and a small favourite against the third. Count the combinations - 6 ways to hold AA, 6 for KK and 16 for AK - and weight each matchup: you win about 39% of the time. That is a call only if the price asks for less, which happens when there is already a lot of money in the pot. See pot odds and expected value.