84s: the 4 combos
A suited hand needs both cards in one suit, so each of the 4 suits gives exactly 1 combo: 4 in all.
Here is every one of the 4 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 84s combos that remain after you hold, or the board shows, one card of each rank.
| Known card | 84s combos left |
|---|---|
| Ace | 4 |
| King | 4 |
| Queen | 4 |
| Jack | 4 |
| Ten | 4 |
| Nine | 4 |
| Eight | 3 |
| Seven | 4 |
| Six | 4 |
| Five | 4 |
| Four | 3 |
| Three | 4 |
| Two | 4 |
Holding an eight or a four takes 84s from 4 combos down to 3, a cut of 25%. Any card of a different rank leaves all 4 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 84s combos left when the board holds a pair of each rank.
| Board pair | 84s combos left |
|---|---|
| Aces | 4 |
| Kings | 4 |
| Queens | 4 |
| Jacks | 4 |
| Tens | 4 |
| Nines | 4 |
| Eights | 2 |
| Sevens | 4 |
| Sixes | 4 |
| Fives | 4 |
| Fours | 2 |
| Threes | 4 |
| Twos | 4 |
Eights or fours paired on the board leave 2 combos of 84s, and no other pair changes the count of 4.
A worked example
You hold an eight and a two. Your opponent's hand is one of 1,225 equally likely two-card holdings. 3 of them are 84s, which is 0.24%, or about 1 in 408. If your two cards had nothing to do with 84s, the same hand would have 4 combos and a 0.33% chance, about 1 in 306.
How strong is 84s?
Measured against a random hand, 84s has 42.70% equity and the number 129 spot of 169, below the middle of the ranking. Wins count once and ties count half.
Together with all stronger classes, 84s covers 966 combos, which is 72.85% of the 1,326 starting hands.
84s 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. 84s makes 22.81% against the top 5 percent (60 combos), 30.39% against the top 10 percent (130 combos), 33.24% against the top 20 percent (264 combos), 34.42% against the top 30 percent (386 combos) and 36.22% against the top 50 percent (658 combos).
| Opponent range | Combos in range | 84s equity |
|---|---|---|
| Top 5 percent | 60 | 22.81% |
| Top 10 percent | 130 | 30.39% |
| Top 20 percent | 264 | 33.24% |
| Top 30 percent | 386 | 34.42% |
| Top 50 percent | 658 | 36.22% |
84s 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 84s an average of 127.00 of them stay possible. That is 3.00 combos fewer than the 130 an opponent has before you look.
Hands next to 84s
Compare 84s with the hand ranked just above it and the hand just below it. 84s has 54.95% equity against 65s, the hand ranked number 128: it wins 52.64% of boards, ties 4.62% and loses 42.74%. It is the favorite in that matchup. 84s has 40.26% equity against 95o, the hand ranked number 130: it wins 38.73% of boards, ties 3.06% and loses 58.21%. It is the underdog in that matchup.
The offsuit twin of 84s is 84o, ranked number 148 with 39.45% equity against a random hand. 84o is 3 times as many combos but 3.26 points weaker against a random hand.
Questions about 84s combos
- How many combos of 84s are there?
- 4 combos, which is 0.30% of the 1,326 two-card starting hands, so you are dealt 84s about 1 time in 331.5.
- How many 84s combos are left if I hold an eight?
- 3 combos of 84s remain out of 4. With a two as your other card, an opponent has it about 1 time in 408.
- Is 84s stronger than 84o?
- Yes, by 3.26 points: 84s has 42.70% equity against a random hand and ranks number 129, while 84o has 39.45% and ranks number 148. 84o has 12 combos to 84s's 4.
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.