Q9s: the 4 combos
There is one queen-nine pairing per suit, and a suited hand counts each suit once, so Q9s is 4 combos.
The list below names all 4 holdings behind Q9s. A hand range that says Q9s 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 | Q9s combos left |
|---|---|
| Ace | 4 |
| King | 4 |
| Queen | 3 |
| 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: a queen 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 Q9s for a board pair of each rank, ace down to two.
| Board pair | Q9s combos left |
|---|---|
| Aces | 4 |
| Kings | 4 |
| Queens | 2 |
| Jacks | 4 |
| Tens | 4 |
| Nines | 2 |
| Eights | 4 |
| Sevens | 4 |
| Sixes | 4 |
| Fives | 4 |
| Fours | 4 |
| Threes | 4 |
| Twos | 4 |
A board pair of queens or nines leaves 2 combos of Q9s; a pair of any other rank leaves all 4.
A worked example
Try it with a queen and a two in your hand. 1,225 two-card holdings are left, 3 of which are Q9s. The chance is 0.24%, about 1 in 408, against 0.33% when none of your cards blocks it.
How strong is Q9s?
By equity against a random hand, Q9s is number 43 of 169, inside the top 60 classes. Its 57.66% 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 Q9s and the list holds 304 combos, 22.93% of the 1,326. The hands above it account for 300 of those.
Q9s against ranges of hands
Opponents rarely hold a random hand, so here is Q9s against five top-percent ranges, each hand weighted by its live combos: 32.33% against the top 5 percent (60 combos), 35.43% against the top 10 percent (130 combos), 37.36% against the top 20 percent (264 combos), 39.80% against the top 30 percent (386 combos) and 47.04% against the top 50 percent (658 combos).
| Opponent range | Combos in range | Q9s equity |
|---|---|---|
| Top 5 percent | 60 | 32.33% |
| Top 10 percent | 130 | 35.43% |
| Top 20 percent | 264 | 37.36% |
| Top 30 percent | 386 | 39.80% |
| Top 50 percent | 658 | 47.04% |
Q9s is behind all five of these ranges, even the top 50 percent, so it is the underdog on average against each of them.
On average 118.00 of the 130 combos in the top 10 percent are still possible once you hold a combo of Q9s. That is 12.00 combos fewer than the 130 an opponent has before you look.
Hands next to Q9s
Hands next to Q9s in the ranking are not always close in a head-to-head. Q9s has 45.69% equity against A6o, the hand ranked number 42: it wins 45.47% of boards, ties 0.43% and loses 54.10%. It is the underdog in that matchup. Q9s has 42.96% equity against K7s, the hand ranked number 44: it wins 42.61% of boards, ties 0.69% and loses 56.70%. It is the underdog in that matchup.
Q9o holds the same two ranks as Q9s but not the same suits, and it ranks number 56 at 55.36% equity. Q9o is 3 times as many combos but 2.30 points weaker against a random hand.
Questions about Q9s combos
- How many combos of Q9s are there?
- 4 combos, which is 0.30% of the 1,326 two-card starting hands, so you are dealt Q9s about 1 time in 331.5.
- How many Q9s combos are left if I hold a queen?
- 3 combos of Q9s remain out of 4. With a two as your other card, an opponent has it about 1 time in 408.
- Is Q9s stronger than Q9o?
- Yes, by 2.30 points: Q9s has 57.66% equity against a random hand and ranks number 43, while Q9o has 55.36% and ranks number 56. Q9o has 12 combos to Q9s's 4.
How this was computed
The count starts from 1,326 two-card hands sorted into 169 classes. Removing the combos that use known cards gives each blocker and board-pair row, with nothing estimated.
Our preflop table counts every board (1,712,304 of them) for each pair of classes, so rank and equity here are exact. Ties are halves; ranges are weighted by live combos.
Source: GeorgeLiPoker's own count of the 52-card deck and its preflop matchup table.