65s: the 4 combos
For 65s, the six and the five must share a suit. With 4 suits and one way to pair them in each, the count is 4.
Every 65s you can be dealt is one of these 4 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 65s, the table lists the combos left after one card of each rank is out of play.
| Known card | 65s combos left |
|---|---|
| Ace | 4 |
| King | 4 |
| Queen | 4 |
| Jack | 4 |
| Ten | 4 |
| Nine | 4 |
| Eight | 4 |
| Seven | 4 |
| Six | 3 |
| Five | 3 |
| Four | 4 |
| Three | 4 |
| Two | 4 |
A six or a five you can see leaves 3 of 4, so 25% of the combos are gone. Any card of a different rank leaves all 4 combos in play.
When the board is paired
A pair on the board uses up two of the four cards of its rank. The 65s combos that survive are in the table, one row per rank.
| Board pair | 65s combos left |
|---|---|
| Aces | 4 |
| Kings | 4 |
| Queens | 4 |
| Jacks | 4 |
| Tens | 4 |
| Nines | 4 |
| Eights | 4 |
| Sevens | 4 |
| Sixes | 2 |
| Fives | 2 |
| Fours | 4 |
| Threes | 4 |
| Twos | 4 |
Pairing sixes or fives on the board cuts 65s to 2 combos. Every other paired board leaves the full 4.
A worked example
Take a hand of a six and a two. Your opponent then holds one of 1,225 possible hands, and 3 of them are 65s. That is 0.24%, or 1 in 408, instead of the 0.33% (1 in 306) you get when your cards do not block it.
How strong is 65s?
Rank 128 of 169 belongs to 65s, below the middle of the ranking, on 43.13% equity against a random hand. Equity here is wins plus half of the ties across all boards.
Add 65s to every hand ranked above it and you hold 962 of the 1,326 two-card hands, 72.55% of all deals.
65s against ranges of hands
A range is a list of hands, and equity against it weights each hand by its live combos. For 65s the five ranges give 24.86% against the top 5 percent (60 combos), 32.58% against the top 10 percent (130 combos), 36.34% against the top 20 percent (264 combos), 37.63% against the top 30 percent (386 combos) and 39.08% against the top 50 percent (658 combos).
| Opponent range | Combos in range | 65s equity |
|---|---|---|
| Top 5 percent | 60 | 24.86% |
| Top 10 percent | 130 | 32.58% |
| Top 20 percent | 264 | 36.34% |
| Top 30 percent | 386 | 37.63% |
| Top 50 percent | 658 | 39.08% |
65s is behind all five of these ranges, even the top 50 percent, so it is the underdog on average against each of them.
Holding 65s removes some premium combos from an opponent's reach: of the 130 in the top 10 percent, an average of 127.00 remain. That is 3.00 combos fewer than the 130 an opponent has before you look.
Hands next to 65s
A one-place difference in rank can be a large or small gap in a direct matchup. 65s has 36.10% equity against 86o, the hand ranked number 127: it wins 29.39% of boards, ties 13.43% and loses 57.18%. It is the underdog in that matchup. 65s has 45.05% equity against 84s, the hand ranked number 129: it wins 42.74% of boards, ties 4.62% and loses 52.64%. It is the underdog in that matchup.
Same ranks, other suits: 65o is the offsuit version, number 145 of 169 at 39.94%. 65o is 3 times as many combos but 3.19 points weaker against a random hand.
Questions about 65s combos
- How many combos of 65s are there?
- 4 combos, which is 0.30% of the 1,326 two-card starting hands, so you are dealt 65s about 1 time in 331.5.
- How many 65s combos are left if I hold a six?
- 3 combos of 65s remain out of 4. With a two as your other card, an opponent has it about 1 time in 408.
- Is 65s stronger than 65o?
- Yes, by 3.19 points: 65s has 43.13% equity against a random hand and ranks number 128, while 65o has 39.94% and ranks number 145. 65o has 12 combos to 65s's 4.
How this was computed
All 1,326 two-card holdings were grouped into classes, and the 65s group was counted. The blocker rows drop each combo containing a visible card; the board-pair rows do it for a visible pair.
Exact matchups, counted over all 1,712,304 boards, supply the rank and equity. A tie counts as half, and ranges are weighted by the combos that remain.
Source: GeorgeLiPoker's own count of the 52-card deck and its preflop matchup table.