K3o: the 12 combos
Pair each of the 4 kings with each of the 4 threes to get 16, then remove the 4 suited pairings. K3o is the other 12.
The list below names all 12 holdings behind K3o. A hand range that says K3o 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 | K3o combos left |
|---|---|
| Ace | 12 |
| King | 9 |
| Queen | 12 |
| Jack | 12 |
| Ten | 12 |
| Nine | 12 |
| Eight | 12 |
| Seven | 12 |
| Six | 12 |
| Five | 12 |
| Four | 12 |
| Three | 9 |
| Two | 12 |
Only two ranks change the count: a king or a three 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 K3o for a board pair of each rank, ace down to two.
| Board pair | K3o combos left |
|---|---|
| Aces | 12 |
| Kings | 6 |
| Queens | 12 |
| Jacks | 12 |
| Tens | 12 |
| Nines | 12 |
| Eights | 12 |
| Sevens | 12 |
| Sixes | 12 |
| Fives | 12 |
| Fours | 12 |
| Threes | 6 |
| Twos | 12 |
A board pair of kings or threes leaves 6 combos of K3o; a pair of any other rank leaves all 12.
A worked example
Try it with a king and a two in your hand. 1,225 two-card holdings are left, 9 of which are K3o. The chance is 0.73%, about 1 in 136, against 0.98% when none of your cards blocks it.
How strong is K3o?
By equity against a random hand, K3o is number 80 of 169, in the middle of the ranking. Its 51.43% 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 K3o and the list holds 592 combos, 44.65% of the 1,326. The hands above it account for 580 of those.
K3o against ranges of hands
Opponents rarely hold a random hand, so here is K3o against five top-percent ranges, each hand weighted by its live combos: 25.46% against the top 5 percent (60 combos), 28.70% against the top 10 percent (130 combos), 32.01% against the top 20 percent (264 combos), 34.03% against the top 30 percent (386 combos) and 41.55% against the top 50 percent (658 combos).
| Opponent range | Combos in range | K3o equity |
|---|---|---|
| Top 5 percent | 60 | 25.46% |
| Top 10 percent | 130 | 28.70% |
| Top 20 percent | 264 | 32.01% |
| Top 30 percent | 386 | 34.03% |
| Top 50 percent | 658 | 41.55% |
K3o 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 121.00 of the 130 combos in the top 10 percent are still possible once you hold a combo of K3o. That is 9.00 combos fewer than the 130 an opponent has before you look.
Hands next to K3o
Hands next to K3o in the ranking are not always close in a head-to-head. K3o has 56.66% equity against J8o, the hand ranked number 79: it wins 56.31% of boards, ties 0.71% and loses 42.98%. It is the favorite in that matchup. K3o has 59.27% equity against Q6o, the hand ranked number 81: it wins 58.92% of boards, ties 0.70% and loses 40.38%. It is the favorite in that matchup.
K3s holds the same two ranks as K3o but not the same suits, and it ranks number 63 at 54.05% equity. K3o has 3 times as many combos as K3s and is 2.63 points weaker against a random hand.
Questions about K3o combos
- How many combos of K3o are there?
- 12 combos, which is 0.90% of the 1,326 two-card starting hands, so you are dealt K3o about 1 time in 110.5.
- How many K3o combos are left if I hold a king?
- 9 combos of K3o remain out of 12. With a two as your other card, an opponent has it about 1 time in 136.
- Is K3o stronger than K3s?
- No: K3o has 51.43% equity against a random hand and ranks number 80, while K3s has 54.05% and ranks number 63. K3s has 4 combos to K3o's 12.
How this was computed
We counted from the deck itself: 1,326 possible two-card hands, each placed in one of 169 classes. A table row then deletes the combos that need a card known to be elsewhere, and the survivors are tallied.
Equity comes from an exact table of preflop matchups, every one of the 1,712,304 boards counted. A tie is half a win, and each hand in a range is weighted by the combos your cards leave.
Source: GeorgeLiPoker's own count of the 52-card deck and its preflop matchup table.