Q5o: the 12 combos
Pair each of the 4 queens with each of the 4 fives to get 16, then remove the 4 suited pairings. Q5o is the other 12.
The list below names all 12 holdings behind Q5o. A hand range that says Q5o means every one of them at once.
How cards you can see change the count
Start from 12 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 | Q5o combos left |
|---|---|
| Ace | 12 |
| King | 12 |
| Queen | 9 |
| Jack | 12 |
| Ten | 12 |
| Nine | 12 |
| Eight | 12 |
| Seven | 12 |
| Six | 12 |
| Five | 9 |
| Four | 12 |
| Three | 12 |
| Two | 12 |
Only two ranks change the count: a queen or a five leaves 9 combos, 25% fewer. Any card of a different rank leaves all 12 combos in play.
When the board is paired
When the board pairs, two more cards are gone. Here is what remains of Q5o for a board pair of each rank, ace down to two.
| Board pair | Q5o combos left |
|---|---|
| Aces | 12 |
| Kings | 12 |
| Queens | 6 |
| Jacks | 12 |
| Tens | 12 |
| Nines | 12 |
| Eights | 12 |
| Sevens | 12 |
| Sixes | 12 |
| Fives | 6 |
| Fours | 12 |
| Threes | 12 |
| Twos | 12 |
A board pair of queens or fives leaves 6 combos of Q5o; a pair of any other rank leaves all 12.
A worked example
Try it with a queen and a two in your hand. 1,225 two-card holdings are left, 9 of which are Q5o. The chance is 0.73%, about 1 in 136, against 0.98% when none of your cards blocks it.
How strong is Q5o?
By equity against a random hand, Q5o is number 89 of 169, in the middle of the ranking. Its 50.12% 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 Q5o and the list holds 654 combos, 49.32% of the 1,326. The hands above it account for 642 of those.
Q5o against ranges of hands
Opponents rarely hold a random hand, so here is Q5o against five top-percent ranges, each hand weighted by its live combos: 24.34% against the top 5 percent (60 combos), 28.46% against the top 10 percent (130 combos), 30.85% against the top 20 percent (264 combos), 32.55% against the top 30 percent (386 combos) and 37.40% against the top 50 percent (658 combos).
| Opponent range | Combos in range | Q5o equity |
|---|---|---|
| Top 5 percent | 60 | 24.34% |
| Top 10 percent | 130 | 28.46% |
| Top 20 percent | 264 | 30.85% |
| Top 30 percent | 386 | 32.55% |
| Top 50 percent | 658 | 37.40% |
Q5o 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 122.00 of the 130 combos in the top 10 percent are still possible once you hold a combo of Q5o. That is 8.00 combos fewer than the 130 an opponent has before you look.
Hands next to Q5o
Hands next to Q5o in the ranking are not always close in a head-to-head. Q5o has 53.23% equity against Q2s, the hand ranked number 88: it wins 32.50% of boards, ties 41.47% and loses 26.03%. It is the favorite in that matchup. Q5o has 70.13% equity against J5s, the hand ranked number 90: it wins 67.64% of boards, ties 4.97% and loses 27.38%. It is the favorite in that matchup.
Q5s holds the same two ranks as Q5o but not the same suits, and it ranks number 72 at 52.77% equity. Q5o has 3 times as many combos as Q5s and is 2.65 points weaker against a random hand.
Questions about Q5o combos
- How many combos of Q5o are there?
- 12 combos, which is 0.90% of the 1,326 two-card starting hands, so you are dealt Q5o about 1 time in 110.5.
- How many Q5o combos are left if I hold a queen?
- 9 combos of Q5o remain out of 12. With a two as your other card, an opponent has it about 1 time in 136.
- Is Q5o stronger than Q5s?
- No: Q5o has 50.12% equity against a random hand and ranks number 89, while Q5s has 52.77% and ranks number 72. Q5s has 4 combos to Q5o's 12.
How this was computed
Counting is done on the deck: 1,326 hands, 169 classes. The rows of each table apply the same rule, delete any combo that uses a card now visible, and report what is left.
For rank and equity, the page reads a table of preflop matchups with all 1,712,304 boards counted per pair of classes. Ties are halves and ranges use live-combo weights.
Source: GeorgeLiPoker's own count of the 52-card deck and its preflop matchup table.