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
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 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. |
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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. |
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UU12 | 78 | A♠K♥ | K♠A♦ | AK | (13 choose 2) = 78 different Ranks 1. Total matchup classes = 78. |
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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. |
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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. |
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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. |
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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. |
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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. |
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PP02 | 0 | A paired hand cannot share 2 ranks. | ||||||||||||

PP01 | 13 | A♠A♥ | A♦A♣ | A | 13 different Ranks 1. Total matchup classes = 13. |
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PP00 | 78 | A♠A♥ | K♦K♣ | AK | (13 choose 2) = 78 different Ranks 1. Total matchup classes = 78. |
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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 | |||

Copyright © 2018 mathematrucker |