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.
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
so
(n! speak n faculty )
Overall there is
different starting hands. From this follows for the probability of two aces
- .
Conditional probability
With fifty-two cards there are four aces in the deck. The probability of getting an ace is therefore included
The probability of receiving an ace in the absence of a card that is an ace is
From this follows a probability of ...
... 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 | 13 x 6 = 78 | 220: 1 | 16: 1 | |||
Same colors |
78 | 78 x 4 = 312 | 331: 1 | 3.25: 1 | |||
Different colors |
78 | 78 x 12 = 936 | 110: 1 | 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
have different starting hands. After the flop, this number goes down
possible hands.
Overall there is heads-up
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
.
Opportunities for the community cards . It follows that it
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
Footnotes
- ↑ 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.