Texas Hold'em odds

Probabilities in Texas Hold'em describes the probable distribution of the chance of winning a given starting hand (hole cards) in the poker game variant Texas Hold'em .

Determination of the probability

The following table roughly applies to the strength of a starting hand. The smaller a number is, the better the hand is. Offsuited hands are on the left or below the main diagonal, hands of the same color ( suited ) are on the right or above the main diagonal.

Pre-flop strategy

	A.	K	Q	J	T	9	8th	7th	6th	5	4th	3	2
A.	1	1	2	2	3	5	5	5	5	5	5	5	5
K	2	1	2	3	4th	6th	7th	7th	7th	7th	7th	7th	7th
Q	3	4th	1	3	4th	5	7th
J	4th	5	5	1	3	4th	6th	8th
T	6th	6th	6th	5	2	4th	5	7th
9	8th	8th	8th	7th	7th	3	4th	5	8th
8th				8th	8th	7th	4th	5	6th	8th
7th							8th	5	5	7th	8th
6th								8th	6th	5	7th
5									8th	6th	6th	8th
4th										8th	7th	7th	8th
3												7th	8th
2													7th

There are basically two ways to determine the probabilities of a starting hand

Result set

Calculate the number of ways that you can hit a particular hand . For example, to get AA , assuming the order is ignored, there are six possibilities, namely A ♠ A ♥ , A ♠ A ♦ , A ♠ A ♣ , A ♥ A ♦ , A ♥ A ♣ , A ♦ A ♣ . The formula for this is

${\displaystyle {\binom {n}{r}}={\frac {n!}{r!\cdot (n-r)!}}}$

so

${\displaystyle {\frac {4!}{(2!\cdot (4-2)!)}}=6}$

(n! speak n faculty )

Overall there is

${\displaystyle {\frac {52\cdot 51}{2!}}={52 \choose 2}=1326}$

different starting hands. From this follows for the probability of two aces

${\displaystyle P={\frac {6}{1326}}={\frac {1}{221}}}$.

Conditional probability

With fifty-two cards there are four aces in the deck. The probability of getting an ace is therefore included

${\displaystyle {\frac {4}{52}}={\frac {1}{13}}}$

The probability of receiving an ace in the absence of a card that is an ace is

${\displaystyle {\frac {3}{51}}={\frac {1}{17}}}$

From this follows a probability of ...

${\displaystyle {\frac {1}{13}}\cdot {\frac {1}{17}}={\frac {1}{221}}}$

... that you get 2 aces when dealing.

Analysis of the starting hands

A total of 1,326 different starting hands are possible in Texas Hold'em. The colors were included in the bill.

From the previous calculations we learned that on average, every 221st hand you get two aces.

Since all colors have the same value in poker, many of the 1,326 possible starting hands are equivalent, at least before the flop . Therefore, hands are basically divided into three groups before the flop

information	Number of hands	Color permutations for each hand	Combinations	Certain hand of the type		Some hand of the guy
				probability	bet	probability	bet
Pocket pair	13	${\displaystyle {4 \choose 2}=6}$	13 x 6 = 78	${\displaystyle {\begin{matrix}{\frac {6}{1326}}\approx 4{,}52\,{}^{0\!}\!/\!_{00}\end{matrix}}}$	220: 1	${\displaystyle {\begin{matrix}{\frac {78}{1326}}\approx 58{,}80\,{}^{0\!}\!/\!_{00}\end{matrix}}}$	16: 1
Same colors	78	${\displaystyle {4 \choose 1}=4}$	78 x 4 = 312	${\displaystyle {\begin{matrix}{\frac {4}{1326}}\approx 3{,}02\,{}^{0\!}\!/\!_{00}\end{matrix}}}$	331: 1	${\displaystyle {\begin{matrix}{\frac {312}{1326}}\approx 235{,}30\,{}^{0\!}\!/\!_{00}\end{matrix}}}$	3.25: 1
Different colors	78	${\displaystyle {4 \choose 1}{3 \choose 1}=12}$	78 x 12 = 936	${\displaystyle {\begin{matrix}{\frac {12}{1326}}\approx 9{,}05\,{}^{0\!}\!/\!_{00}\end{matrix}}}$	110: 1	${\displaystyle {\begin{matrix}{\frac {936}{1326}}\approx 705{,}90\,{}^{0\!}\!/\!_{00}\end{matrix}}}$	0.417: 1

The following are the probabilities for certain hands:

hand	probability	bet
AK s or two other specific suiteds	0.302%	331: 1
AA or some other specific pair	0.452%	220: 1
AK s, KQ s, QJ s, or JT s	1.207%	81.9: 1
AK or some other specific non-couple	1.207%	81.9: 1
AA , KK , or QQ	1.357%	72.7: 1
AA , KK , QQ or JJ	1.810%	54.3: 1
Cards of the same color, J or higher	1.810%	54.3: 1
AA , KK , QQ , YY , or DD	2.262%	43.2: 1
Cards of the same color, T or higher	3.107%	32.2: 1
Suited connectors	3.922%	24.5: 1
Connectors, T or better	4.827%	19.7: 1
Two cards, Q or higher	4.977%	19.1: 1
Any couple	5.882%	16: 1
Two cards, J or higher	9.050%	10.1: 1
Two cards, T or higher	14.329%	5.98: 1
Connectors	15.686%	5.38: 1
Two cards, 9 or higher	20.815%	3.81: 1
Neither connected nor suited, at least one card 9 or lower	53.394%	0.873: 1

Starting hands in heads-up

In heads-up , the opposing player can

${\displaystyle {\frac {50\cdot 49}{2}}={50 \choose 2}=1225}$

have different starting hands. After the flop, this number goes down

${\displaystyle {\frac {47\cdot 46}{2}}={47 \choose 2}=1081}$

possible hands.

Overall there is heads-up

${\displaystyle {52 \choose 2}{50 \choose 2}=1.624.350}$

different ways of confronting which cards the players have in hand. We now assume that two players hold their hands until after the river and we see a showdown like this. There are

${\displaystyle {48 \choose 5}=1.712.304}$.

Opportunities for the community cards . It follows that it

${\displaystyle 1.712.304\cdot 1.624.350=3.679.075.400={52 \choose 9}}$

So there are around 3.68 billion opportunities for distributing community and hole cards .

Comparison of two starting hands

The following table contains the probabilities for the outcome of a clash between the starting hands of two players

Favorite against underdog	probability	bet
Couple against undercards	83.0%	4.9: 1
Pair versus lower pair	82.0%	4.5: 1
Pair against an overcard and an undercard	71.0%	2.5: 1
2 over against 2 under cards	63.0%	1.7: 1
Pair against 2 overcards	55.0%	1.2: 1

These numbers cannot be given exactly, after all, the colors of the cards can also influence the result.

Example: A ♠ A ♣ wins against K ♠ Q ♣ by 87.650% (0.490% for the split pot ), against 6 ♦ 7 ♦ but only by 76.81% (0.32% for the split pot).

Web links

• Several tables of odds in Texas Holdem

Footnotes

1. ↑ From a purely mathematical point of view, it makes no difference whether more players played at the beginning, but who put their cards away (both cards that were discarded and not dealt are equally disregarded in the calculation). But of course the opponents would only have thrown a bad hand in the game . It is implicitly assumed here that there were only two players from the start (definition 1. of heads-up ) and that it is a completely new hand.