A9s: the 4 combos
There is one ace-nine pairing per suit, and a suited hand counts each suit once, so A9s is 4 combos.
The list below names all 4 holdings behind A9s. A hand range that says A9s means every one of them at once.
How cards you can see change the count
Start from 4 combos and remove every combo that uses a card you can see. The table does that for one known card of each rank.
| Known card | A9s combos left |
|---|---|
| Ace | 3 |
| King | 4 |
| Queen | 4 |
| Jack | 4 |
| Ten | 4 |
| Nine | 3 |
| Eight | 4 |
| Seven | 4 |
| Six | 4 |
| Five | 4 |
| Four | 4 |
| Three | 4 |
| Two | 4 |
Only two ranks change the count: an ace or a nine leaves 3 combos, 25% fewer. Any card of a different rank leaves all 4 combos in play.
When the board is paired
When the board pairs, two more cards are gone. Here is what remains of A9s for a board pair of each rank, ace down to two.
| Board pair | A9s combos left |
|---|---|
| Aces | 2 |
| Kings | 4 |
| Queens | 4 |
| Jacks | 4 |
| Tens | 4 |
| Nines | 2 |
| Eights | 4 |
| Sevens | 4 |
| Sixes | 4 |
| Fives | 4 |
| Fours | 4 |
| Threes | 4 |
| Twos | 4 |
A board pair of aces or nines leaves 2 combos of A9s; a pair of any other rank leaves all 4.
A worked example
Try it with an ace and a two in your hand. 1,225 two-card holdings are left, 3 of which are A9s. The chance is 0.24%, about 1 in 408, against 0.33% when none of your cards blocks it.
How strong is A9s?
By equity against a random hand, A9s is number 18 of 169, inside the top 30 classes. Its 62.78% counts wins in full and ties as half, over all 1,712,304 boards, for every pair of hands that can be dealt.
Stop the ranking at A9s and the list holds 114 combos, 8.60% of the 1,326. The hands above it account for 110 of those.
A9s against ranges of hands
Opponents rarely hold a random hand, so here is A9s against five top-percent ranges, each hand weighted by its live combos: 33.57% against the top 5 percent (60 combos), 35.61% against the top 10 percent (130 combos), 49.06% against the top 20 percent (264 combos), 54.26% against the top 30 percent (386 combos) and 58.71% against the top 50 percent (658 combos).
| Opponent range | Combos in range | A9s equity |
|---|---|---|
| Top 5 percent | 60 | 33.57% |
| Top 10 percent | 130 | 35.61% |
| Top 20 percent | 264 | 49.06% |
| Top 30 percent | 386 | 54.26% |
| Top 50 percent | 658 | 58.71% |
The line sits between the top 20 and top 30 percent: A9s is behind the narrower range and ahead of the wider one.
On average 107.00 of the 130 combos in the top 10 percent are still possible once you hold a combo of A9s. That is 23.00 combos fewer than the 130 an opponent has before you look.
Hands next to A9s
Hands next to A9s in the ranking are not always close in a head-to-head. A9s has 46.76% equity against 66, the hand ranked number 17: it wins 46.55% of boards, ties 0.42% and loses 53.03%. It is the underdog in that matchup. A9s has 32.14% equity against ATo, the hand ranked number 19: it wins 27.08% of boards, ties 10.11% and loses 62.81%. It is the underdog in that matchup.
A9o holds the same two ranks as A9s but not the same suits, and it ranks number 25 at 60.77% equity. A9o is 3 times as many combos but 2.01 points weaker against a random hand.
Questions about A9s combos
- How many combos of A9s are there?
- 4 combos, which is 0.30% of the 1,326 two-card starting hands, so you are dealt A9s about 1 time in 331.5.
- How many A9s combos are left if I hold an ace?
- 3 combos of A9s remain out of 4. With a two as your other card, an opponent has it about 1 time in 408.
- Is A9s stronger than A9o?
- Yes, by 2.01 points: A9s has 62.78% equity against a random hand and ranks number 18, while A9o has 60.77% and ranks number 25. A9o has 12 combos to A9s's 4.
How this was computed
We counted from the deck itself: 1,326 possible two-card hands, each placed in one of 169 classes. A table row then deletes the combos that need a card known to be elsewhere, and the survivors are tallied.
Equity comes from an exact table of preflop matchups, every one of the 1,712,304 boards counted. A tie is half a win, and each hand in a range is weighted by the combos your cards leave.
Source: GeorgeLiPoker's own count of the 52-card deck and its preflop matchup table.