52s combos

52s: the 4 combos

Fix the suit, and the five and two of that suit are the only choice; four suits make 4 combos.

These are the 4 specific hands that count as 52s. Each tile is one two-card holding, and no two tiles are the same hand.

• 5♠2♠
• 5♥2♥
• 5♦2♦
• 5♣2♣

How cards you can see change the count

This is the blocker table for 52s. One visible card of a given rank removes the combos that need it, and the rest stay possible.

With a five or a two in view, 3 of the 4 combos of 52s remain, which removes 25%. Any card of a different rank leaves all 4 combos in play.

When the board is paired

Boards with a pair remove two cards of one rank at once. The rows give the 52s combos left for each paired rank.

Only a board pair of fives or twos matters here: 2 combos of 52s remain, against 4 for any other pair.

A worked example

Suppose you look down at a five and a three. Any opponent hand comes from 1,225 possibilities. 52s makes up 3 of them, which is 0.24% (1 in 408); if you held neither card's rank in common with 52s, it would be 4 of them, 0.33%.

How strong is 52s?

Order all 169 classes by equity against a random hand and 52s lands at number 155, among the 30 weakest classes, with 37.85%. A tie is worth half a win in that figure.

Down to and including 52s, the ranking holds 1,182 combos, or 89.14% of deals; 1,178 of them belong to hands above it.

52s against ranges of hands

Five ranges, five equities for 52s. Counting only the combos your cards leave possible, it has 22.33% against the top 5 percent (60 combos), 29.73% against the top 10 percent (130 combos), 33.27% against the top 20 percent (264 combos), 34.34% against the top 30 percent (386 combos) and 35.28% against the top 50 percent (658 combos).

52s is behind all five of these ranges, even the top 50 percent, so it is the underdog on average against each of them.

Blockers work on ranges as well as hands. Against the 130 combos of the top 10 percent, 52s leaves an average of 130.00. That is no loss at all: 52s blocks none of those hands.

Hands next to 52s

Look at 52s against the hands that rank next to it. 52s has 43.86% equity against 64o, the hand ranked number 154: it wins 37.60% of boards, ties 12.52% and loses 49.88%. It is the underdog in that matchup. 52s has 39.17% equity against 62s, the hand ranked number 156: it wins 27.83% of boards, ties 22.69% and loses 49.48%. It is the underdog in that matchup.

52o is what the same two ranks look like when they are offsuit: 12 combos at number 166, 34.28% equity. 52o is 3 times as many combos but 3.56 points weaker against a random hand.

Questions about 52s combos

How many combos of 52s are there?

4 combos, which is 0.30% of the 1,326 two-card starting hands, so you are dealt 52s about 1 time in 331.5.

How many 52s combos are left if I hold a five?

3 combos of 52s remain out of 4. With a three as your other card, an opponent has it about 1 time in 408.

Is 52s stronger than 52o?

Yes, by 3.56 points: 52s has 37.85% equity against a random hand and ranks number 155, while 52o has 34.28% and ranks number 166. 52o has 12 combos to 52s's 4.

How this was computed

All 1,326 two-card holdings were grouped into classes, and the 52s 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.