Poker probability (Texas hold 'em)

In poker, the probability of many events can be determined by direct calculation. This article discusses how to compute the probabilities for many commonly occurring events in the game of Texas hold 'em and provides some probabilities and oddsref_label|odds|1|^ for specific situations. In most cases, the probabilities and odds are approximations due to rounding.

When calculating probabilities for a card game such as Texas Hold 'em, there are two basic approaches. the first approach is to determine the number of outcomes that satisfy the condition being evaluated and divide this by the total number of possible outcomes.For example, there are six outcomes (ignoring order) for being dealt a pair of aces in Hold' em: {A♣, A♥}, {A♠, A♦}, {A♠, A♣}, {A♥, A♦}, {A♥, A♠}, and {A♦, A♣}. There are 52 ways to pick the first card and 51 ways to pick the second card and two ways to order the two cards yielding (52×51)/2=1326 possible outcomes when being dealt two cards (also ignoring order).This gives a probability of being dealt two aces of

The second approach is to use conditional probabilities, or in more complex situations, a decision tree. There are 4 ways to be dealt an ace out of 52 choices for the first card resulting in a probability of There are 3 ways of getting dealt an ace out of 51 choices on the second card after being dealt an ace on the first card for a probability of The conditional probability of being dealt two aces is the product of the two probabilities: (Note that in this case the total is not divided by 2 ways of ordering the cards because both cards must be an ace. Reordering would still require the first and second cards to be an ace, so there is only one way to order the two cards.)

Often, the key to determining probability is selecting the best approach for a given problem. This article uses both of these approaches.

Starting hands

The probability of being dealt various starting hands can be explicitly calculated. In Texas Hold 'em, a player is dealt two down (or "hole" or "pocket") cards. The first card can be any one of 52 playing cards in the deck and the second card can be any one of the 51 remaining cards. This gives 52 × 51 ÷ 2 = 1,326 possible starting hand combinations. (Since the order of the cards is not significant, the 2,652 permutations are divided by the 2 ways of ordering two cards.) Alternatively, the number of possible starting hands is represented as the binomial coefficient

:$\{52\; choose\; 2\}\; =\; 1,326$

which is the number of possible combinations of choosing 2 cards from a deck of 52 playing cards.

The 1,326 starting hands can be reduced for purposes of determining the probability of starting hands for Hold 'em—since suits have no relative value in poker, many of these hands are identical in value before the flop. The only factors determining the strength of a starting hand are the ranks of the cards and whether the cards share the same suit. Of the 1,326 combinations, there are 169 distinct starting hands grouped into three "shapes:" 13 "pocket pairs" (paired hole cards), 13 × 12 ÷ 2 = 78 "suited hands" and 78 "unsuited hands;" 13 + 78 + 78 = 169. The relative probability of being dealt a hand of each given shape is different. The following shows the probabilities and odds of being dealt each type of starting hand.

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These odds are general approximations only derived from averaging all of the hand matchups in each category. The actual head-to-head probabilities for any two starting hands vary depending on a number of factors, including:
* Suited or unsuited starting hands;
* Shared suits between starting hands;
* Connectedness of non-pair starting hands;
* Proximity of card ranks between the starting hands (lowering straight potential);
* Proximity of card ranks toward A or 2 (lowering straight potential);
* Possibility of split pot.

For example, A♠ A♣ vs. K♠ Q♣ is 87.65% to win (0.49% to split), but A♠ A♣ vs. 7♦ 6♦ is 76.81% to win (0.32% to split).

The mathematics for computing all of the possible matchups is quite complex. However, a computer program can perform a brute force evaluation of the 1,712,304 possible boards for any given pair of starting hands in seconds.

Starting hands against multiple opponents

When facing two opponents, for any given starting hand the number of possible combinations of hands the opponents can have is

:$\{50\; choose\; 2\}\{48\; choose\; 2\}\; =\; 1,381,800$

hands. For calculating probabilities we can ignore the distinction between the two opponents holding A♠ J♥ and 8♥ 8♣ and the opponents holding 8♥ 8♣ and A♠ J♥. The number of ways that hands can be distributed between $n$ opponents is $n!$ (the factorial of n). So the number of unique hand combinations $H$ against two opponents is

:$H\; =\; \{50\; choose\; 2\}\{48\; choose\; 2\}\; div\; 2!\; =\; 690,900,$

and against three opponents is

:$H\; =\; \{50\; choose\; 2\}\{48\; choose\; 2\}\{46\; choose\; 2\}\; div\; 3!\; =\; 238,360,500,$

and against $n$ opponents is

:$H\; =\; prod\_\{k=1\}^n\; \{52\; -\; 2k\; choose\; 2\}\; div\; k!,$ or alternatively $H\; =\; \{50\; choose\; 2n\}\; imes\; (2n-1)!!,$

It is important to note here that x!! is not the same as (x!)!, click the link just below to see the calculation

where $(2n-1)!!$ (!! is the double factorial operator) is the number of ways to distribute $2n$ cards between $n$ hands of two cards each.ref_label|prod|3|^ The following table shows the number of hand combinations for up to nine opponents.

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The following table gives the probability that a hand is facing two or more larger pairs before the flop. From the previous equations, the probability $P\_m$ is computed as

:$P\_m\; =\; P\_2\; +\; P\_3\; +\; cdots\; +\; P\_n.$

:

The flop

The value of a starting hand can change dramatically after the flop. Regardless of initial strength, any hand can flop the nuts—for example, if the flop comes with three 2s, any hand holding the fourth 2 has the nuts (though additional cards could still give another player a higher four of a kind or a straight flush). Conversely, the flop can undermine the perceived strength of any hand—a player holding A♣ A♥ would not be happy to see 8♠ 9♠ 10♠ on the flop because of the straight and flush possibilities.

There are

:$\{50\; choose\; 3\}\; =\; 19,600$

possible flops for any given starting hand. By the turn the total number of combinations has increased to

:$\{50\; choose\; 4\}\; =\; 230,300$

and on the river there are

:$\{50\; choose\; 5\}\; =\; 2,118,760$

possible boards to go with the hand.

The following are some general probabilities about what can occur on the board. These assume a "random" starting hand for the player.

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* Note: When drawing to a full house or four of a kind with a pocket pair that has hit "trips" (three of a kind) on the flop, there are 6 outs to get a full house by pairing the board and one out to make four of a kind. This means that if the turn does not pair the board or make four of a kind, there will be 3 additional outs on the river, for a total of 10, to pair the turn card and make a full house. This makes the probability of drawing to a full house or four of a kind on the turn or river 0.334 and the odds are 1.99 : 1. This makes drawing to a full house or four of a kind by the river about 8½ outs.

It is worth noting in the preceding table that if a player doesn't fold before the river, a hand with at least 14 outs after the flop has a better than 50% chance to catch one of its outs by the river. With 20 or more outs, a hand is a better than 2 : 1 favorite to catch at least one out by the river.

"See the article on pot odds for examples of how these probabilities might be used in gameplay decisions."

Estimating probability of drawing outs

Few poker players have the mathematical ability to calculate odds in the middle of a poker hand. One solution is to memorize the actual odds of drawing outs after the flop and after the turn, since these odds are needed frequently for making decisions.

Another solution some players use is an easily-calculated approximation of the probability for drawing outs.

Approximating odds after the flop

With two cards to come, the percent chance of hitting one of "x" outs is about 4x. This approximation gives roughly accurate probabilities for up to about 12 outs, with an absolute average error of 0.9, a maximum absolute error of 3, a relative average error of 3.5, and a maximum relative error of 6.8.

A slightly more complicated, but significantly more accurate approximation of drawing outs after the flop is to use 4x only for 1 to 9 outs, and (3x+9) for 10 or more outs. This approximation has a maximum absolute error of less than 1 for 1 to 19 outs and maximum relative error of less than 5 for 2 to 23 outs.

Approximating odds after the turn

With one card to come, the percent chance of hitting one of "x" cards is about 2x. This approximation has a constant relative error of an 8 underestimation, which produces a linearly increasing absolute error of about 1 for each 6 outs.

A more accurate approximation is (2x+(2x÷10). This is easily done by first multiplying "x" by 2, then rounding the result to the nearest multiple of ten, and adding the 10's digit to the first result. For example, to calculate the odds of hitting one of 12 outs on the river: 12 × 2 = 24, 24 rounds to 20, so the approximation is 24 + 2 = 26. This approximation has a maximum absolute error of less than 0.9 for 1 to 19 outs and a maximum relative error of 3.5 for more than 3 outs.

Chart: post-flop odds approximations

The following shows the approximations and their absolute and relative errors for both methods of approximation.

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The preceding table assumes the following definitions.

;Outside straight and straight flush:Drawing to a sequence of three cards of consecutive rank from 3-4-5 to 10-J-Q where two cards can be added to either end of the sequence to make a straight or straight flush.

;Inside+outside straight and straight flush:Drawing to a straight or straight flush where one required rank can be combined with one of two other ranks to make the hand. This includes sequences like 5-7-8 which requires a 6 plus either a 4 or 9 as well as the sequences J-Q-K, which requires a 10 plus either a 9 or A, and 2-3-4 which requires a 5 plus either an A or 6.

;Inside-only straight and straight flush:Drawing to a straight or straight flush where there are only two ranks that make the hand. This includes hands such as 5-7-9 which requires a 6 and an 8 as well as A-2-3 which requires a 4 and a 5.

Compound outs

The strongest runner-runner probabilities lie with hands that are drawing to multiple hands with different runner-runner combinations. These include hands that can make a straight, flush or straight flush, as well as four of a kind or a full house. Calculating these probabilities requires adding the compound probabilities for the various outs, taking care to account for any shared hands. For example, if $P\_s$ is the probability of a runner-runner straight, $P\_f$ is the probability of a runner-runner flush, and $P\_\{sf\}$ is the probability of a runner-runner straight flush, then the compound probability $P$ of getting one of these hands is

:$P\; =\; P\_s\; +\; P\_f\; -\; P\_\{sf\}.$

The probability of the straight flush is subtracted from the total because it is already included in both the probability of a straight and the probability of a flush, so it has been added twice and must therefore be subtracted from the compound outs of a straight or flush.

The following table gives the compound probability and odds of making a runner-runner for common situations and the equivalent normal outs.

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Some hands have even more runner-runner chances to improve. For example, holding the hand J♠ Q♠ after a flop of 10♠ J♥ 7♦ there are several runner-runner hands to make at least a straight. The hand can get two cards from the common outs of {J, Q} (5 cards) to make a full house or four of a kind, can get a J (2 cards) plus either a 7 or 10 (6 cards) to make a full house from these independent disjoint outs, and is drawing to the compound outs of a flush, outside straight or straight flush. The hand can also make {7, 7} or {10, 10} (each drawing from 3 common outs) to make a full house, although this will make four of a kind for anyone holding the remaining 7 or 10 or a bigger full house for anyone holding an overpair. Working from the probabilities from the previous tables and equations, the probability $P$ of making one of these runner-runner hands is a compound probability

:$P\; =\; 0.08326\; +\; 0.00925\; +\; frac\{2\; imes\; 6\}\{1081\}\; +\; (0.00278\; imes\; 2)\; approx\; 0.1092$

and odds of 8.16 : 1 for the equivalent of 2.59 normal outs. Almost all of these runner-runners give a winning hand against an opponent who had flopped a straight holding 8, 9,ref_label|runner|4|^ but only some give a winning hand against A♠ 2♠ (this hand makes bigger flushes when a flush is hit) or against K♣ Q♦ (this hand makes bigger straights when a straight is hit with 8 9). When counting outs, it is necessary to adjust for which outs are likely to give a winning hand—this is where the skill in poker becomes more important than being able to calculate the probabilities.

See also

Poker topics:
* Texas hold 'em
* Poker probability
* Texas hold 'em hands
* Poker strategy
* Pot odds
* Poker

Math and probability topics:
* Probability
* Odds
* Sample space
* Event (probability theory)
* Binomial coefficient
* Combination
* Permutation
* Combinatorial game theory
* Game complexity
* Set theory
* Gaming mathematics

Notes

# The odds presented in this article use the notation x : 1 which translates to "x to 1 odds against" the event happening. The odds are calculated from the probability "p" of the event happening using the formula: odds = [(1 − "p") ÷ "p"] : 1, or odds = [(1 ÷ "p") − 1] : 1. Another way of expressing the odds x : 1 is to state that there is a "1 in x+1" chance of the event occurring or the probability of the event occurring is 1/(x+1). So for example, the odds of a role of a fair six-sided die coming up three is 5 : 1 against because there are 5 chances for a number other than three and 1 chance for a three; alternatively, this could be described as a 1 in 6 chance or probability of a three being rolled because the three is 1 of 6 equally-likely possible outcomes.
# ^ note_label|pruning|2|anote_label|pruning|2|b By removing reflection and applying aggressive search tree pruning, it is possible to reduce the number of unique head-to-head hand combinations from 207,025 to less than 50,000. Reflection eliminates redundant calculations by observing that given hands $h\_1$ and $h\_2$, if $w\_1$ is the probability of $h\_1$ beating $h\_2$ in a showdown and $s$ is the probability of $h\_1$ splitting the pot with $h\_2$, then the probability $w\_2$ of $h\_2$ beating $h\_1$ is $w\_2\; =\; 1\; -\; (s\; +\; w\_1)$, thus eliminating the need to evaluate $h\_2$ against $h\_1$. Pruning is possible, for example, by observing that Q♥ J♥ has the same chance of winning against both 8♦ 7♣ and 8♦ 7♠ (but "not" the same probability as against 8♥ 7♣ because sharing the heart affects the flush possibilities for each hand).
# See "Capital Pi notation for multiplication" for a description of the $prod$ (capital π or pi) symbol.
# In the example, if the opponent is holding either 8♥ 9♥ or 8♦ 9♦, then the opponent wins with a flush if the player makes a straight using two hearts or two diamonds, respectively. If the opponent is holding 8♦ 9♦, then the opponent wins with a straight flush if the player makes a full house with 10♦ J♦.

References

*cite book | author = Mike Petriv
year = 1996
title = Hold'em Odds Book
publisher = Objective Observer Press
id = ISBN 0-9681223-0-2
*cite book | author = King Yao
year = 2005
title = Weighing the Odds in Hold 'em Poker
publisher = Pi Yee Press
id = ISBN 0-935926-25-9
*cite book | author = Dan Harrington, Bill Robertie
year = 2005
title = Harrington on Hold'em Volume 1: Strategic Play
publisher = Two Plus Two Publishing
id = ISBN 1-880685-33-7

External links

* [http://www.math.sfu.ca/~alspach/computations.html Poker Computations] by Brian Alspach
* [http://www.tightpoker.com/poker_odds.html Calculating Probability Odds in Texas Hold'em]
* [http://www.cardschat.com/odds-for-dummies.php Poker Odds for Dummies]
* [http://probability.infarom.ro/holdempoker.html Applied Probability in Texas Hold'em]