76o: the 12 combos
For 76o, there are 4 sevens and 4 sixes, so 16 pairings, and the 4 that share a suit belong to the suited hand. That leaves 12.
Every 76o you can be dealt is one of these 12 holdings. They are listed once each, higher card first.
How cards you can see change the count
Known cards cannot be in an opponent's hand. For 76o, the table lists the combos left after one card of each rank is out of play.
| Known card | 76o combos left |
|---|---|
| Ace | 12 |
| King | 12 |
| Queen | 12 |
| Jack | 12 |
| Ten | 12 |
| Nine | 12 |
| Eight | 12 |
| Seven | 9 |
| Six | 9 |
| Five | 12 |
| Four | 12 |
| Three | 12 |
| Two | 12 |
A seven or a six you can see leaves 9 of 12, so 25% of the combos are gone. Any card of a different rank leaves all 12 combos in play.
When the board is paired
A pair on the board uses up two of the four cards of its rank. The 76o combos that survive are in the table, one row per rank.
| Board pair | 76o combos left |
|---|---|
| Aces | 12 |
| Kings | 12 |
| Queens | 12 |
| Jacks | 12 |
| Tens | 12 |
| Nines | 12 |
| Eights | 12 |
| Sevens | 6 |
| Sixes | 6 |
| Fives | 12 |
| Fours | 12 |
| Threes | 12 |
| Twos | 12 |
Pairing sevens or sixes on the board cuts 76o to 6 combos. Every other paired board leaves the full 12.
A worked example
Take a hand of a seven and a two. Your opponent then holds one of 1,225 possible hands, and 9 of them are 76o. That is 0.73%, or 1 in 136, instead of the 0.98% (1 in 102) you get when your cards do not block it.
How strong is 76o?
Rank 133 of 169 belongs to 76o, below the middle of the ranking, on 42.32% equity against a random hand. Equity here is wins plus half of the ties across all boards.
Add 76o to every hand ranked above it and you hold 1,006 of the 1,326 two-card hands, 75.87% of all deals.
76o against ranges of hands
A range is a list of hands, and equity against it weights each hand by its live combos. For 76o the five ranges give 20.94% against the top 5 percent (60 combos), 29.60% against the top 10 percent (130 combos), 33.16% against the top 20 percent (264 combos), 34.99% against the top 30 percent (386 combos) and 36.49% against the top 50 percent (658 combos).
| Opponent range | Combos in range | 76o equity |
|---|---|---|
| Top 5 percent | 60 | 20.94% |
| Top 10 percent | 130 | 29.60% |
| Top 20 percent | 264 | 33.16% |
| Top 30 percent | 386 | 34.99% |
| Top 50 percent | 658 | 36.49% |
76o is behind all five of these ranges, even the top 50 percent, so it is the underdog on average against each of them.
Holding 76o removes some premium combos from an opponent's reach: of the 130 in the top 10 percent, an average of 124.00 remain. That is 6.00 combos fewer than the 130 an opponent has before you look.
Hands next to 76o
A one-place difference in rank can be a large or small gap in a direct matchup. 76o has 44.18% equity against 92s, the hand ranked number 132: it wins 42.67% of boards, ties 3.02% and loses 54.31%. It is the underdog in that matchup. 76o has 57.15% equity against 74s, the hand ranked number 134: it wins 40.49% of boards, ties 33.31% and loses 26.20%. It is the favorite in that matchup.
Same ranks, other suits: 76s is the suited version, number 115 of 169 at 45.37%. 76o has 3 times as many combos as 76s and is 3.05 points weaker against a random hand.
Questions about 76o combos
- How many combos of 76o are there?
- 12 combos, which is 0.90% of the 1,326 two-card starting hands, so you are dealt 76o about 1 time in 110.5.
- How many 76o combos are left if I hold a seven?
- 9 combos of 76o remain out of 12. With a two as your other card, an opponent has it about 1 time in 136.
- Is 76o stronger than 76s?
- No: 76o has 42.32% equity against a random hand and ranks number 133, while 76s has 45.37% and ranks number 115. 76s has 4 combos to 76o's 12.
How this was computed
Combos are counted by listing all 1,326 two-card hands in a 52-card deck and sorting each into its class. The known-card rows remove every combo that uses a card of that rank; the board-pair rows remove those that use either of two such cards.
Rank and equity read our table of all 169 by 169 preflop matchups, each counted over every one of the 1,712,304 boards. Ties count as half a win, and a range is weighted by its live combos.
Source: GeorgeLiPoker's own count of the 52-card deck and its preflop matchup table.