Module 4 · Lesson 21 of 39 · Free
Domination: why A9 loses to AK
A♠ 9♦ against A♥ K♣, all in before the flop: the ace-nine has 26.1% equity, about one pot in four. Both hands hold an ace, so the ace decides nothing; the other card, the kicker, decides, and the king beats the nine. A hand that shares a card with a better hand is dominated.
The hand
All in before the flop, both hands face up: your A♠ 9♦ against the Tight-aggressive player's A♥ K♣. All five board cards are still to come.
Predict first
Which card on the board would help your A♠ 9♦ most?
- An ace: you would have a pair of aces
- A nine
- A king: a big card for your ace
Show the answer
Answer: A nine
An ace gives him a pair of aces too, and his king still beats your nine. A king gives him a pair of kings. A nine gives you a pair he does not have, and three nines are left.
Working it out
Count the cards. Of the 48 cards neither of you holds, 3 nines help you, 3 kings help him, and the 2 aces left help neither. At least one nine comes on 1 − C(45, 5) ÷ C(48, 5) = 28.6% of boards, but some of those boards bring him a king as well, and a few boards with no nine give you a straight or a flush. Counting all 1,712,304 boards, A♠ 9♦ wins 23.8% of the time and splits the pot 4.6%: 23.8% + 4.6% ÷ 2 = 26.1%. Take away the shared ace and the same hand is the favorite: against K♥ Q♣ it has 58.0%.
Domination works with any shared card: K♠ Q♦ against K♥ J♣ has 73.6%, and a pair dominates a hand that holds one of its cards: K♠ K♥ against A♦ K♣ has 70.0%. A bigger pair against a smaller one is even more one-sided, about four times in five: K♠ K♥ against 7♦ 7♣, 79.8%. The other kinds of matchups are milder. A pair against two higher cards is close to a coin flip: 7♠ 7♥ against A♦ K♣, 55.4% for the pair. Against two higher cards that are suited and connected, a small pair can even be behind: 5♠ 5♦ against J♥ T♥, 47.5%. Two higher cards against two lower ones are the favorite: A♠ K♥ against 8♦ 7♦, 58.1%.
Formula
Equity = (wins + splits ÷ 2) ÷ boards: (407,694 + 79,513 ÷ 2) ÷ 1,712,304 = 26.1%
Check your understanding
Q♠ J♦ against Q♥ T♣, all in before the flop. Which is true?
- Q♠ J♦ dominates Q♥ T♣
- It is a coin flip: both hold a queen
- Neither: a queen on the board decides it
Show the answer
Answer: Q♠ J♦ dominates Q♥ T♣
Both hands hold a queen, so the other cards decide, and the jack beats the ten. Counting every board, Q♠ J♦ has 72.4%.
T♠ T♦ against A♥ K♣, all in before the flop. Who is the favorite?
- T♠ T♦, by a little
- A♥ K♣: it has two higher cards
- T♠ T♦, by about four to one
Show the answer
Answer: T♠ T♦, by a little
A pair against two higher cards is close to a coin flip: counting every board, T♠ T♦ has 57.3% and A♥ K♣ 42.7%.
Your 6♠ 6♥ against his 9♦ 9♣, all in before the flop. You need one of the two sixes left. What is the chance that at least one of them comes among the five board cards, in %?
Show the answer
Answer: 19.9%
Of the 48 cards neither of you holds, 2 are sixes: 1 − C(46, 5) ÷ C(48, 5) = 1 − 80.1% = 19.9%.