mathematrucker
Equity/Tie % Tables
SUITED  x -connectors vs
ALL  y -connectors
UNSUITED  x -connectors vs
UNSUITED  x -connectors
UNSUITED  y -connectors vs
ALL  x -connectors
0 1 2 3 4 5 6 7 8 9 10 11
0 0 0 0 0 0 0 0 0 0 0
0 1 1 1 1 1 1 1 1 1 1 1
1 1 2 2 2 2 2 2 2 2 2 2
2 2 2 3 3 3 3 3 3 3 3 3
3 3 3 3 4 4 4 4 4 4 4 4
4 4 4 4 4 5 5 5 5 5 5 5
5 5 5 5 5 5 6 6 6 6 6 6
6 6 6 6 6 6 6 7 7 7 7 7
7 7 7 7 7 7 7 7 8 8 8 8
8 8 8 8 8 8 8 8 8 9 9 9
9 9 9 9 9 9 9 9 9 9 10 10
10 10 10 10 10 10 10 10 10 10 10 11
11 11 11 11 11 11 11 11 11 11 11 11
0 1 2 3 4 5 6 7 8 9 10 11
Win % By Hand
J♠T♠ VERSUS 5♥5♦
52% < EQUITY > 48%
HIGH CARD
0.00
0.00
ONE PAIR
14.76
8.58
TWO PAIR
17.32
16.10
TRIPS
2.51
9.92
STRAIGHT
8.81
1.68
FLUSH
6.06
2.11
FULL HOUSE
2.28
7.86
QUADS 0.13 0.89
STR FLUSH 0.22 0.03
Unique Texas Hold'em Matchups:
Preflop High Hand Equity and Tie Percentages
Many heads up matchups are essentially identical preflop. This is because poker does not rank suits.

For example, even though there are six different pocket aces versus pocket kings matchups with all four suits present, all six of these are mathematically the same. As such they belong to the same matchup class, or collection of mathematically identical matchups. Moreover these six form a complete matchup class, since every other pocket aces versus pocket kings matchup involves at least one shared suit, the presence of which alters the balance of flushes and straight flushes.

To count how many unique matchups exist in Texas hold'em, we proceed to single out a representative matchup from each matchup class and then count up all the representatives.

It is helpful to begin by selecting suit distributions for every possible matchup class.

Hand Type 1 Hand Type 2 Shared Suits Hand 1 Hand 2
SUITED SUITED 1 ♠♠ ♠♠
SUITED SUITED 0 ♠♠ ♥♥
SUITED UNSUITED 1 ♠♠ ♠♥
SUITED UNSUITED 0 ♠♠ ♥♦
UNSUITED UNSUITED 2 ♠♥ ♠♥
UNSUITED UNSUITED 1 ♠♥ ♠♦
UNSUITED UNSUITED 0 ♠♥ ♦♣

Every representative matchup will have its suits assigned from the above table.

Every Texas hold'em starting hand has exactly one shape: suited (S), unsuited (U), or pair (P). These three shapes give rise to six different matchup types: suited vs. suited (SS), suited vs. unsuited (SU), suited vs. pair (SP), unsuited vs. unsuited (UU), unsuited vs. pair (UP), and pair vs. pair (PP). Since every hand has exactly one shape, every matchup has exactly one type.

Every matchup has 0, 1, or 2 shared suits and 0, 1, or 2 shared ranks. Each matchup type therefore breaks into nine mutually exclusive subcategories based on how many shared suits and shared ranks there are. Every matchup belongs to exactly one of these subcategories, but some subcategories do not contain any matchups (for example no matchup involving a pair can have 2 shared ranks).

The table below displays all 54 subcategories and the number of distinct matchup classes within each. Each subcategory label gives the number of shared suits and shared ranks (in that order). Since the suits are predetermined, our only task is to count the different possible rank combinations.

The Hand 1 and Hand 2 columns give example matchups. The Ranks 1 column gives the initial rank(s) selected and the Ranks 2 column gives the rest. Explanations appear in the Notes column at the right.

Subcat Matchup Classes Hand 1 Hand 2 Ranks 1 Ranks 2 Notes
SS22 0 A suited hand cannot share 2 suits.
SS21 0 A suited hand cannot share 2 suits.
SS20 0 A suited hand cannot share 2 suits.
SS12 0 A suited hand cannot share both a suit and a rank with another suited hand.
SS11 0 A suited hand cannot share both a suit and a rank with another suited hand.
SS10 2,145 A♠K♠ Q♠J♠ AK QJ (13 choose 2) = 78 different Ranks 1.
(11 choose 2) = 55 different Ranks 2.
Must divide by 2 to avoid double counting.
Total matchup classes = (78 × 55) / 2 = 2,145.
SS02 78 A♠K♠ A♥K♥ AK (13 choose 2) = 78 different Ranks 1.
Total matchup classes = 78.
SS01 858 A♠K♠ A♥Q♥ A KQ 13 different Ranks 1.
(12 choose 2) = 66 different Ranks 2.
Total matchup classes = 13 × 66 = 858.
SS00 2,145 A♠K♠ Q♥J♥ AK QJ (13 choose 2) = 78 different Ranks 1.
(11 choose 2) = 55 different Ranks 2.
Must divide by 2 to avoid double counting.
Total matchup classes = (78 × 55) / 2 = 2,145.
SU22 0 A suited hand cannot share 2 suits.
SU21 0 A suited hand cannot share 2 suits.
SU20 0 A suited hand cannot share 2 suits.
SU12 0 A suited hand cannot share both of its ranks with any hand it shares a suit with.
SU11 1,716 A♠K♠
A♠Q♠		 Q♠A♥
K♠A♥		 A KQ 13 different Ranks 1.
(12 choose 2) = 66 different Ranks 2.
Must multiply by 2 to account for asymmetry.
Total matchup classes = (13 × 66) × 2 = 1,716.
SU10 8,580 A♠K♠
A♠K♠		 Q♠J♥
J♠Q♥		 AK QJ (13 choose 2) = 78 different Ranks 1.
(11 choose 2) = 55 different Ranks 2.
Must multiply by 2 to account for asymmetry.
Total matchup classes = (78 × 55) × 2 = 8,580.
SU02 78 A♠K♠ A♥K♦ AK (13 choose 2) = 78 different Ranks 1.
Total matchup classes = 78.
SU01 1,716 A♠K♠
A♠Q♠		 A♥Q♦
A♥K♦		 A KQ 13 different Ranks 1.
(12 choose 2) = 66 different Ranks 2.
Must multiply by 2 to account for asymmetry.
Total matchup classes = (13 × 66) × 2 = 1,716.
SU00 4,290 A♠K♠ Q♥J♦ AK QJ (13 choose 2) = 78 different Ranks 1.
(11 choose 2) = 55 different Ranks 2.
Total matchup classes = 78 × 55 = 4,290.
SP22 0 A suited hand cannot share 2 suits.
SP21 0 A suited hand cannot share 2 suits.
SP20 0 A suited hand cannot share 2 suits.
SP12 0 A paired hand cannot share 2 ranks.
SP11 0 A suited hand cannot share both a suit and a rank with a paired hand.
SP10 858 K♠Q♠ A♠A♥ A KQ 13 different Ranks 1.
(12 choose 2) = 66 different Ranks 2.
Total matchup classes = 13 × 66 = 858.
SP02 0 A paired hand cannot share 2 ranks.
SP01 156 A♠K♠ A♥A♦ A K 13 different Ranks 1.
12 different Ranks 2.
Total matchup classes = 13 × 12 = 156.
SP00 858 K♠Q♠ A♥A♦ A KQ 13 different Ranks 1.
(12 choose 2) = 66 different Ranks 2.
Total matchup classes = 13 × 66 = 858.
UU22 78 A♠K♥ K♠A♥ AK (13 choose 2) = 78 different Ranks 1.
Total matchup classes = 78.
UU21 858 A♠K♥ Q♠A♥ A KQ 13 different Ranks 1.
(12 choose 2) = 66 different Ranks 2.
Total matchup classes = 13 × 66 = 858.
UU20 4,290 A♠Q♥
A♠J♥		 K♠J♥
K♠Q♥		 AK QJ (13 choose 2) = 78 different Ranks 1.
(11 choose 2) = 55 different Ranks 2.
Must divide by 2 to avoid double counting.
Must multiply by 2 to account for asymmetry.
Total matchup classes = 78 × 55 = 4,290.
UU12 78 A♠K♥ K♠A♦ AK (13 choose 2) = 78 different Ranks 1.
Total matchup classes = 78.
UU11 2,574 A♠K♥
A♠Q♥
K♠A♥		 Q♠A♦
K♠A♦
Q♠A♦		 A KQ 13 different Ranks 1.
(12 choose 2) = 66 different Ranks 2.
Must multiply by 3 to account for asymmetries.
Total matchup classes = (13 × 66) × 3 = 2,574.
UU10 8,580 A♠Q♥
A♠J♥		 K♠J♦
K♠Q♦		 AK QJ (13 choose 2) = 78 different Ranks 1.
(11 choose 2) = 55 different Ranks 2.
Must multiply by 2 to account for asymmetry.
Total matchup classes = (78 × 55) × 2 = 8,580.
UU02 78 A♠K♥ A♦K♣ AK (13 choose 2) = 78 different Ranks 1.
Total matchup classes = 78.
UU01 858 A♠K♥ A♦Q♣ A KQ 13 different Ranks 1.
(12 choose 2) = 66 different Ranks 2.
Total matchup classes = 13 × 66 = 858.
UU00 2,145 A♠K♥ Q♦J♣ AK QJ (13 choose 2) = 78 different Ranks 1.
(11 choose 2) = 55 different Ranks 2.
Must divide by 2 to avoid double counting.
Total matchup classes = (78 × 55) / 2 = 2,145.
UP22 0 A paired hand cannot share 2 ranks.
UP21 0 A paired hand cannot share its rank and both suits.
UP20 858 K♠Q♥ A♠A♥ A KQ 13 different Ranks 1.
(12 choose 2) = 66 different Ranks 2.
Total matchup classes = 13 × 66 = 858.
UP12 0 A paired hand cannot share 2 ranks.
UP11 156 K♠A♥ A♠A♦ A K 13 different Ranks 1.
12 different Ranks 2.
Total matchup classes = 13 × 12 = 156.
UP10 1,716 K♠Q♥
Q♠K♥		 A♠A♦
A♠A♦		 A KQ 13 different Ranks 1.
(12 choose 2) = 66 different Ranks 2.
Must multiply by 2 to account for asymmetry.
Total matchup classes = (13 × 66) × 2 = 1,716.
UP02 0 A paired hand cannot share 2 ranks.
UP01 156 A♠K♥ A♦A♣ A K 13 different Ranks 1.
12 different Ranks 2.
Total matchup classes = 13 × 12 = 156.
UP00 858 K♠Q♥ A♦A♣ A KQ 13 different Ranks 1.
(12 choose 2) = 66 different Ranks 2.
Total matchup classes = 13 × 66 = 858.
PP22 0 A paired hand cannot share 2 ranks.
PP21 0 A paired hand cannot share its rank and both suits.
PP20 78 A♠A♥ K♠K♥ AK (13 choose 2) = 78 different Ranks 1.
Total matchup classes = 78.
PP12 0 A paired hand cannot share 2 ranks.
PP11 0 A paired hand cannot share both a suit and its rank with another paired hand.
PP10 78 A♠A♥ K♠K♦ AK (13 choose 2) = 78 different Ranks 1.
Total matchup classes = 78.
PP02 0 A paired hand cannot share 2 ranks.
PP01 13 A♠A♥ A♦A♣ A 13 different Ranks 1.
Total matchup classes = 13.
PP00 78 A♠A♥ K♦K♣ AK (13 choose 2) = 78 different Ranks 1.
Total matchup classes = 78.

Here is the grand total with subtotals by matchup type.

Matchup Type Matchup Classes
SS 5,226
SU 16,380
SP 1,872
UU 19,539
UP 3,744
PP 247
Total: 47,008


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