K3s combos

K3s: the 4 combos

There is one king-three pairing per suit, and a suited hand counts each suit once, so K3s is 4 combos.

All 4 ways to hold K3s are shown here, one tile each. Counting them is how a range gets its size.

• K♠3♠
• K♥3♥
• K♦3♦
• K♣3♣

How cards you can see change the count

Every card you can see is a card nobody else can hold. The table shows how many of the 4 combos of K3s are still possible when you hold one card of each rank, and the same count applies when that card is on the board.

Only two ranks change the count: a king or a three leaves 3 combos, 25% fewer. Any card of a different rank leaves all 4 combos in play.

When the board is paired

How does a paired board change the count? For K3s, the rows below show the combos left after the board pairs each rank in turn.

A board pair of kings or threes leaves 2 combos of K3s; a pair of any other rank leaves all 4.

A worked example

Worked example: your cards are a king and a two, so 1,225 two-card hands remain for your opponent. Of those, 3 are K3s, a 0.24% chance (about 1 in 408). With two cards that do not touch K3s, it would be 4 combos and 0.33%.

How strong is K3s?

K3s holds rank 63 among the 169 classes, in the middle of the ranking. Against a random hand it has 54.05% equity: wins in full, ties as half, every board counted.

454 combos rank above K3s, and with its own 4 the running total is 458, 34.54% of the 1,326 hands.

K3s against ranges of hands

Take the strongest hands in rank order and ask how K3s does against them. The answers, with live combos as weights: 29.20% against the top 5 percent (60 combos), 32.50% against the top 10 percent (130 combos), 35.70% against the top 20 percent (264 combos), 37.65% against the top 30 percent (386 combos) and 44.76% against the top 50 percent (658 combos).

K3s is behind all five of these ranges, even the top 50 percent, so it is the underdog on average against each of them.

Your cards matter to the opponent's range too: with K3s, 121.00 of the 130 top-10-percent combos survive on average. That is 9.00 combos fewer than the 130 an opponent has before you look.

Hands next to K3s

Two quick matchups show where K3s stands among its neighbours. K3s has 43.65% equity against K6o, the hand ranked number 62: it wins 26.63% of boards, ties 34.04% and loses 39.34%. It is the underdog in that matchup. K3s has 54.16% equity against T9s, the hand ranked number 64: it wins 53.80% of boards, ties 0.72% and loses 45.48%. It is the favorite in that matchup.

Next to K3s sits K3o, its offsuit twin: rank 80, 12 combos, 51.43% equity against a random hand. K3o is 3 times as many combos but 2.63 points weaker against a random hand.

Questions about K3s combos

How many combos of K3s are there?

4 combos, which is 0.30% of the 1,326 two-card starting hands, so you are dealt K3s about 1 time in 331.5.

How many K3s combos are left if I hold a king?

3 combos of K3s remain out of 4. With a two as your other card, an opponent has it about 1 time in 408.

Is K3s stronger than K3o?

Yes, by 2.63 points: K3s has 54.05% equity against a random hand and ranks number 63, while K3o has 51.43% and ranks number 80. K3o has 12 combos to K3s's 4.

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.