Poker probability

:"See also Poker probability (Texas hold 'em) and Poker probability (Omaha) for probabilities specific to those games."

In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands.

Frequency of 5-card poker hands

The following enumerates the frequency of each hand, given all combinations of 5 cards randomly drawn from a full deck of 52 without replacement. Wild cards are not considered. The probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand by the total number of 5-card hands (the sample space, five-card hands). The odds are defined as the ratio "(1/p) - 1 : 1", where "p" is the probability. Note that the cumulative column contains the probability of being dealt that hand or any of the hands ranked higher than it. (The frequencies given are exact; the probabilities and odds are approximate.)

Please note, that in the interests of calculating these values for yourselves, the function "nCr" on most scientific calculators can be used. To see what the actual formula looks like, please see the "Any five card poker hand" below.

As can be seen from the table, just over half the time a player gets a hand that has no pairs, three- or four-of-a-kinds. (50.7%)

If aces are not low, simply rotate the hand descriptions so that 6-high replaces 5-high for the best hand and ace-high replaces king-high as the worst hand.

Derivation of frequencies for 5-card lowball hands

The following computations show how the above frequencies for 5-card lowball poker hands were determined. To understand these derivations, the reader should be familiar with the basic properties of the binomial coefficients and their interpretation as the number of ways of choosing elements from a given set. See also: sample space and event (probability theory).

The probability for any specific low hand with 5 distinct ranks (i.e. no paired cards) is the same. The frequency of a 5-high hand or any a specific low hand is calculated by making 5 independent choices for the suit for each rank, which is:

:$\{4\; choose\; 1\}^5\; =\; 1,024$

There is one way to choose the ranks for a five-high hand:

:$\{5\; choose\; 5\}\; =\; 1$

To determine the number of distinct six-high hands, once the six is chosen, the other 4 ranks are chosen from the 5 ranks A to 5, which is:

:$\{5\; choose\; 4\}\; =\; 5$

This can be generalized for any non-paired low hand. Where $r$ is the highest rank in the hand (numbering jack–king as 11–13), the number of distinct low hands is:

:$D\; =\; \{r-1\; choose\; 4\}$

and the frequency of low hands that are $r$-high is $1,024\; cdot\; D$.

Derivation for lowball hands without straights and flushes:

In the case where straights and flushes count against a low hand, the frequency of a specific hand must subtract the 4 combinations of suits that yield a flush, and the calculation for the number of distinct hands must subtract the $r\; -\; 4$ combinations of ranks that yield a straight. This gives the following frequency for low hands of rank $r$ that do not include a straight or a flush:

:$left\; [\; \{4\; choose\; 1\}^5\; -\; 4\; ight]\; cdot\; left\; [\; \{r-1\; choose\; 4\}\; -\; (r\; -\; 4)\; ight]$

Frequency of 7-card lowball poker hands

:"See "Rank of hands (poker)#Low-poker ranking" for a more complete discussion of lowball poker hands."

In some variants of poker a player uses the best five-card low hand made from seven cards. In most variants of lowball, the ace is counted as the lowest card and straights and flushes don't count against a low hand, so the lowest hand is the five-high hand A-2-3-4-5, also called a "wheel". The probability is calculated based on , the total number of 7-card combinations. (The frequencies given are exact; the probabilities and odds are approximate.)

:

As can be seen from the table, 95.4% of the time a player can make a 5-card low hand that has no pairs or three-of-a-kind.

If aces are not low, simply rotate the hand descriptions so that 6-high replaces 5-high for the best hand and ace-high replaces king-high as the worst hand.

Derivation of frequencies for 7-card lowball hands

The following computations show how the above frequencies for 7-card lowball poker hands were determined. To understand these derivations, the reader should be familiar with the basic properties of the binomial coefficients and their interpretation as the number of ways of choosing elements from a given set. See also: sample space and event (probability theory).

To make a low hand of a specific rank four ranks are chosen that are lower than the high rank. Where $r$ is the highest rank in the hand (numbering jack–king as 11–13), the number of sets of 5 ranks that can make a low hand is:

:$\{r-1\; choose\; 4\}$

There are then three different ways to choose the remaining two cards that are not used in the low hand. Each of these cases must be considered separately:

7 distinct ranks

In this type of hand the two additional ranks are chosen from the ranks higher than $r$, so this type of hand can only occur when there are at least two ranks greater than $r$—that is, jack-high or better hands. The suits can be assigned by making 7 independent choices for the suit for each rank, so the number of ways to make a low hand with two distinct higher ranks is:

:$\{13-r\; choose\; 2\}\{4\; choose\; 1\}^7\; =\; \{13-r\; choose\; 2\}\; cdot\; 16,384$

6 distinct ranks

In this type of hand there are 6 distinct ranks and one pair. The additional rank is chosen from the ranks higher than $r$, so this type of hand can only occur when there is at least one rank greater than $r$—that is, queen-high or better hands. One of the 6 ranks is chosen for the pair and two of the four cards in that rank are chosen. The suits for the remaining 5 ranks are assigned by making 5 independent choices for each rank, so the number of ways to make a low hand with one higher ranks and a pair is:

:$\{13-r\; choose\; 1\}\{6\; choose\; 1\}\{4\; choose\; 2\}\{4\; choose\; 1\}^5\; =\; \{13-r\; choose\; 1\}\; cdot\; 36,864$

5 distinct ranks

There are two ways to choose 5 distinct ranks for seven cards. Either two pair and three unpaired ranks or three of a kind and four unpaired ranks.

:Two pair

:In this type of hand there are 5 distinct ranks and two pair. Two of the 5 ranks are chosen for the pairs and two of the four cards in each rank are chosen. The suits for the remaining 3 ranks are assigned by making 3 independent choices for each rank, so the number of ways to make a low hand with two pair is:

::$\{5\; choose\; 2\}\{4\; choose\; 2\}^2\{4\; choose\; 1\}^3\; =\; 23,040$

:Three of a kind

:In this type of hand there are 5 distinct ranks and three of a kind. One of the 5 ranks is chosen for the three of a kind and three of the four cards in the rank are chosen. The suits for the remaining 4 ranks are assigned by making 4 independent choices for each rank, so the number of ways to make a low hand with three of a kind is:

::$\{5\; choose\; 1\}\{4\; choose\; 3\}\{4\; choose\; 1\}^4\; =\; 5,120$

Thus there are $23,040\; +\; 5,120\; =\; 28,160$ ways to make a low hand with five distinct ranks.

Derivation

Thus where $r$ is a rank from 5 to jack (11), the total number of $r$-high low hands is:

:$\{r-1\; choose\; 4\}\; cdot\; left\; [\; \{13-r\; choose\; 2\}\; cdot\; 16,384\; +\; \{13-r\; choose\; 1\}\; cdot\; 36,864\; +\; 28,160\; ight]$

The total number of queen-high low hands is:

:$\{11\; choose\; 4\}\; cdot\; left\; [\; \{1\; choose\; 1\}\; cdot\; 36,864\; +\; 28,160\; ight]\; =\; \{11\; choose\; 4\}\; cdot\; 65,024\; =\; 21,457,920$

The total number of king-high low hands is:

:$\{12\; choose\; 4\}\; cdot\; 28,160\; =\; 13,939,200$

See also

Poker topics:
* Poker probability (Texas hold 'em)
* Poker probability (Omaha)
* 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

External links

* [http://www.math.sfu.ca/~alspach/ Brian Alspach's mathematics and poker page]
* [http://mathworld.wolfram.com/Poker.html MathWorld: Poker]
* [http://www.probabilityof.com/poker.shtml Poker probabilities including conditional calculations]
* [http://www.suffecool.net/poker/table1.html Numerous poker probability tables]
* [http://www.durangobill.com/Poker.html 5, 6, and 7 card poker probabilities]
* [http://www.suffecool.net/poker/7462.html The 7,462 and 4,824 equivalence classes]
* [http://www.compatiblepoker.com/Poker+Odds+Table.cms.htm Preflop, After Flop and Chance of Making Hand Odds]
* [http://www.world-of-poker.org/pokertip/holdem-odds-outs.htm Odds and Outs probability table]