Q3o: the 12 combos
An offsuit hand pairs any queen with any three of a different suit: 4 x 4 = 16 pairings, minus the 4 suited ones, leaves 12.
Here is every one of the 12 combos, written as the two cards you would see. Clubs, diamonds, hearts and spades each appear in the way the count above requires.
How cards you can see change the count
Seeing a card changes which hands are possible. Here is the count of Q3o combos that remain after you hold, or the board shows, one card of each rank.
| Known card | Q3o combos left |
|---|---|
| Ace | 12 |
| King | 12 |
| Queen | 9 |
| Jack | 12 |
| Ten | 12 |
| Nine | 12 |
| Eight | 12 |
| Seven | 12 |
| Six | 12 |
| Five | 12 |
| Four | 12 |
| Three | 9 |
| Two | 12 |
Holding a queen or a three takes Q3o from 12 combos down to 9, a cut of 25%. Any card of a different rank leaves all 12 combos in play.
When the board is paired
A paired board shows two cards of one rank, and those two cards cannot be in an opponent's hand either. The table shows the Q3o combos left when the board holds a pair of each rank.
| Board pair | Q3o combos left |
|---|---|
| Aces | 12 |
| Kings | 12 |
| Queens | 6 |
| Jacks | 12 |
| Tens | 12 |
| Nines | 12 |
| Eights | 12 |
| Sevens | 12 |
| Sixes | 12 |
| Fives | 12 |
| Fours | 12 |
| Threes | 6 |
| Twos | 12 |
Queens or threes paired on the board leave 6 combos of Q3o, and no other pair changes the count of 12.
A worked example
You hold a queen and a two. Your opponent's hand is one of 1,225 equally likely two-card holdings. 9 of them are Q3o, which is 0.73%, or about 1 in 136. If your two cards had nothing to do with Q3o, the same hand would have 12 combos and a 0.98% chance, about 1 in 102.
How strong is Q3o?
Measured against a random hand, Q3o has 48.22% equity and the number 98 spot of 169, in the middle of the ranking. Wins count once and ties count half.
Together with all stronger classes, Q3o covers 722 combos, which is 54.45% of the 1,326 starting hands.
Q3o against ranges of hands
Against a range, equity is averaged over every hand in it, weighted by how many combos of each hand are still possible. Q3o makes 23.62% against the top 5 percent (60 combos), 27.76% against the top 10 percent (130 combos), 30.07% against the top 20 percent (264 combos), 31.47% against the top 30 percent (386 combos) and 35.99% against the top 50 percent (658 combos).
| Opponent range | Combos in range | Q3o equity |
|---|---|---|
| Top 5 percent | 60 | 23.62% |
| Top 10 percent | 130 | 27.76% |
| Top 20 percent | 264 | 30.07% |
| Top 30 percent | 386 | 31.47% |
| Top 50 percent | 658 | 35.99% |
Q3o is behind all five of these ranges, even the top 50 percent, so it is the underdog on average against each of them.
Your own cards block part of an opponent's strong range. The top 10 percent holds 130 combos, and when you hold a combo of Q3o an average of 122.00 of them stay possible. That is 8.00 combos fewer than the 130 an opponent has before you look.
Hands next to Q3o
Compare Q3o with the hand ranked just above it and the hand just below it. Q3o has 70.43% equity against J3s, the hand ranked number 97: it wins 68.34% of boards, ties 4.20% and loses 27.47%. It is the favorite in that matchup. Q3o has 54.30% equity against 98o, the hand ranked number 99: it wins 53.82% of boards, ties 0.96% and loses 45.22%. It is the favorite in that matchup.
The suited twin of Q3o is Q3s, ranked number 82 with 51.02% equity against a random hand. Q3o has 3 times as many combos as Q3s and is 2.80 points weaker against a random hand.
Questions about Q3o combos
- How many combos of Q3o are there?
- 12 combos, which is 0.90% of the 1,326 two-card starting hands, so you are dealt Q3o about 1 time in 110.5.
- How many Q3o combos are left if I hold a queen?
- 9 combos of Q3o remain out of 12. With a two as your other card, an opponent has it about 1 time in 136.
- Is Q3o stronger than Q3s?
- No: Q3o has 48.22% equity against a random hand and ranks number 98, while Q3s has 51.02% and ranks number 82. Q3s has 4 combos to Q3o's 12.
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.