Dominated hands: why ace-nine struggles against ace-king

Ace-nine is an ace, and an ace feels like a strong card. Against a random hand it is fine. Against ace-king it is in trouble, and the reason is not that the nine is weak but that the ace is shared. Poker players call this being dominated: your best card is matched by your opponent's, so it cannot win you the hand, and their second card is bigger than yours.

What domination costs

I dealt random boards for each matchup below (800,000 per line, so the figures are good to about a tenth of a point) with the suits fixed so the hands share none. Equity counts a chop as half a win.

Ace-king offsuit against ace-nine offsuit comes out at 71.6% to win, 4.6% to chop and 23.8% to lose. Ace-nine therefore holds 23.8% + (4.6% ÷ 2) = 26.1% of the pot, and ace-king 73.9%. That is nearly three to one.

The comparison that shows what is going on is the first pair of bars. Ace-nine against king-queen has no shared rank and is a 58% to 42% favourite, simply because it has the ace. Put the same ace-nine against ace-king and it falls from 58% to 26%, a drop of 32 points. Nothing about ace-nine changed. The opponent's hand did.

King-queen is dominated the same way by ace-king, because the king is shared. It holds 25.8% (about 26%). In Holdem Coach: Poker Trainer the "Which hand is in front?" lesson shows two hands and asks you to choose the one more likely to win, which is the question this table answers.

Where the 26% comes from

With both hands dealt, 48 cards are unseen, and the board takes five of them: C(48,5) = 1,712,304 possible boards. Three kings and three nines remain, plus two aces.

1. A king comes, no nine: ace-king pairs its second card and ace-nine does not. The count is C(45,5) − C(42,5) = 371,091 boards, or 21.7%.
2. A nine comes, no king: now ace-nine has the pair. By the same count it is also 371,091 boards, 21.7%. The nine is no rarer than the king.
3. An ace comes: both hands pair aces and the kicker decides. Ace-king's king beats the nine. This happens on 1,712,304 − 1,370,754 = 341,550 boards, 19.9%, and it only goes the other way if a nine also arrives and no king does.

So the nine is as likely to appear as the king, and yet ace-nine wins just 23.8%. The difference is the ace. Even the cleanest case for ace-nine, a nine with no king and no ace on the board, covers C(43,5) − C(40,5) = 304,590 boards, or 17.8%. When an ace arrives the kicker decides, and when a king arrives with the nine the king pairs ace-king too. Either way the better kicker collects.

The lower card hardly matters

You might expect a closer kicker to help. It does not, much. I ran ace-king offsuit against ace-queen, ace-ten, ace-nine and ace-deuce, all offsuit and sharing no suit with it. The dominated hand held between 26 and 27% every time.

That is the useful number to remember. Once you share the ace with a hand holding a better kicker, you start at about 26%, and the nine is worth barely more than a deuce. Domination is about the shared rank, not how good the second card is.

Suit helps by the usual amount, about four points here. Ace-nine suited in hearts against ace-king offsuit with no heart in common came out at 30.4% (2,000,000 boards dealt). That is a better 26%, but still a long way behind, and the suited or offsuit post shows why the suit alone does not rescue a hand.

What it means for the call

Take the price. A half-pot bet asks you to win 25% of the time and a pot-sized bet asks for 33.3%, as worked out in the pot odds post. Against exactly ace-king, ace-nine's 26.1% scrapes past the first and falls well short of the second.

That is the optimistic reading. The bet in front of you comes from a range of hands, and the 26% is the share before any more betting. If the board then gives you an ace, you are likely to be the one paying: you both hold top pair, the opponent's king outkicks your nine, and the extra bets go in on the wrong side. Players call this reverse implied odds. When ace-nine does win, it often wins a small pot, and when it loses it loses a big one.

Using it at the table

1. Be careful with ace-small. The higher the opponent's kicker, the more of the hands that dominate you they can hold. Position changes how often that happens; see why position matters.
2. Check the shared card first. If you hold A9 and someone raises, the hands that hurt you most are the ones holding an ace with a kicker above a nine.
3. Prefer hands that dominate. Ace-king sits on the right side of every other ace hand, so it is the one that does the dominating.
4. Compare with a hand that does not share a rank. The 58% against king-queen shows how much of the damage is domination; for another matchup worked out board by board, see pair against two overcards.

The hand charts in the app show at a glance which two-card hands to play from each seat, and the "Which hands to play" lesson asks whether your two cards are worth raising with before anyone has bet. Ace-nine is a good one to try both on.