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Poker probability

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.

History

People thought about probability and gambling long before the invention of poker. Gambling led to the development of probability theory in the late 1400’s. When playing a game with high stakes, players wanted to know what the chance of winning would be. In 1494, Fra Luca Paccioli released his work Summa de arithmetica, geometria, proportioni e proportionalita which was the first written text on probability. Motivated by Paccioli’s work, Girolamo Cardano (1501-1576) made further developments in probability theory. His work from 1550, titled Liber de Ludo Aleae, discussed the concepts of probability and how they directly related to gambling. However, his work did not receive any recognition because it was not published until after his death. Blaise Pascal (1623-1662) also contributed to probability theory. His friend, Chevalier de Méré, was an avid gambler with the goal to become wealthy from it. De Méré tried a new mathematical approach to a gambling game but did not get the desired results. Determined to know why his strategy was unsuccessful, he consulted with Pascal. Pascal’s work on this problem began an important correspondence between him and fellow mathematician Pierre de Fermat (1601-1665). Communicating through letters, the two continued to exchange their ideas and thoughts. These interactions led to the conception of basic probability theory. To this day, many gamblers still rely on the basic concepts of probability theory in order to make informed decisions while gambling.

Frequency of 5-card poker hands

The following chart enumerates the (absolute) 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. In this chart:

The nCr function on most scientific calculators can be used to calculate hand frequencies; entering nCr with 52 and 5, for example, yields as above.

Hand Distinct Hands Frequency Probability Cumulative probability Odds Mathematical expression of absolute frequency
Royal flush

 1 4 0.000154% 0.000154% 649,739 : 1
Straight flush (excluding royal flush)

 9 36 0.00139% 0.0014% 72,192 : 1
Four of a kind

 156 624 0.0240% 0.0256% 4,164 : 1
Full house

 156 3,744 0.1441% 0.17% 693 : 1
Flush (excluding royal flush and straight flush)

 1,277 5,108 0.1965% 0.367% 508 : 1
Straight (excluding royal flush and straight flush)

 10 10,200 0.3925% 0.76% 254 : 1
Three of a kind

 858 54,912 2.1128% 2.87% 46.3 : 1
Two pair

 858 123,552 4.7539% 7.62% 20.0 : 1
One pair

 2,860 1,098,240 42.2569% 49.9% 1.37 : 1
No pair / High card

 1,277 1,302,540 50.1177% 100% 0.995 : 1
Total 7,462 2,598,960 100% --- 0 : 1

The royal flush is a case of the straight flush. It can be formed 4 ways (one for each suit), giving it a probability of 0.000154% and odds of 649,739 : 1.

When ace-low straights and ace-low straight flushes are not counted, the probabilities of each are reduced: straights and straight flushes each become 9/10 as common as they otherwise would be. The 4 missed straight flushes become flushes and the 1,020 missed straights become no pair.

Note that since suits have no relative value in poker, two hands can be considered identical if one hand can be transformed into the other by swapping suits. For example, the hand 3♣ 7♣ 8♣ Q♠ A♠ is identical to 3♦ 7♦ 8♦ Q♥ A♥ because replacing all of the clubs in the first hand with diamonds and all of the spades with hearts produces the second hand. So eliminating identical hands that ignore relative suit values, there are only 134,459 distinct hands.

The number of distinct poker hands is even smaller. For example, 3♣ 7♣ 8♣ Q♠ A♠ and 3♦ 7♣ 8♦ Q♥ A♥ are not identical hands when just ignoring suit assignments because one hand has three suits, while the other hand has only two - that difference could affect the relative value of each hand when there are more cards to come. However, even though the hands are not identical from that perspective, they still form equivalent poker hands because each hand is an A-Q-8-7-3 high card hand. There are 7,462 distinct poker hands.

Derivation of frequencies of 5-card poker hands

The following computations show how the above frequencies for 5-card 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).

Frequency of 7-card poker hands

In some popular variations of poker, a player uses the best five-card poker hand out of seven cards. The frequencies are calculated in a manner similar to that shown for 5-card hands, except additional complications arise due to the extra two cards in the 7-card poker hand. The total number of distinct 7-card hands is . It is notable that the probability of a no-pair hand is less than the probability of a one-pair or two-pair hand.

The Ace-high straight flush or royal flush is slightly more frequent (4324) than the lower straight flushes (4140 each) because the remaining two cards can have any value; a King-high straight flush, for example, cannot have the Ace of its suit in the hand (as that would make it ace-high instead).

Hand Frequency Probability Cumulative Odds
Royal flush 4,324 0.0032% 0.0032% 30,939 : 1
Straight flush (excl. royal flush) 37,260 0.0279% 0.0311% 3,589.6 : 1
Four of a kind 224,848 0.168% 0.199% 594 : 1
Full house 3,473,184 2.60% 2.80% 37.5 : 1
Flush 4,047,644 3.03% 5.82% 32.1 : 1
Straight 6,180,020 4.62% 10.4% 20.6 : 1
Three of a kind 6,461,620 4.83% 15.3% 19.7 : 1
Two pair 31,433,400 23.5% 38.8% 3.26 : 1
One pair 58,627,800 43.8% 82.6% 1.28 : 1
No pair 23,294,460 17.4% 100% 4.74 : 1
Total 133,784,560 100% 100% 0 : 1

(The frequencies given are exact; the probabilities and odds are approximate.)

Since suits have no relative value in poker, two hands can be considered identical if one hand can be transformed into the other by swapping suits. Eliminating identical hands that ignore relative suit values leaves 6,009,159 distinct 7-card hands.

The number of distinct 5-card poker hands that are possible from 7 cards is 4,824. Perhaps surprisingly, this is fewer than the number of 5-card poker hands from 5 cards because some 5-card hands are impossible with 7 cards (e.g. 7-high).

Derivation of frequencies of 7-card poker hands

See "7-Card Poker Hands" by Brian Alspach for the article on which this explanation is based.

The following computations show how the above frequencies for 7-card 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).

1 triple, 1 pair and 2 kickers
The triple may be 1 of 13 ranks, and by definition 3 of the 4 of that rank are chosen. The pair may be 1 of the remaining 12 ranks, and (again, by definition) 2 of the 4 of that rank are chosen. The ranks of the 2 kickers are chosen from the remaining 11 ranks, and 1 of the 4 of each rank are chosen. Thus, the total number of full houses in this form is:
1 triple and 2 pairs
The triple is chosen the same way as before, the ranks of the two pairs are chosen from the remaining 12 ranks, and the 2 of the 4 of each rank are chosen as usual. Thus, the total number of full houses in this form is:
2 triples and 1 kicker
The ranks of both triples are chosen from the 13, then the rank of the kicker is chosen from the remaining 11 ranks. Thus, the total number of full houses in this form is:
Thus, the total number of full houses is:
7 distinct ranks
In this type of straight, all 7 cards are of unique ranks (i.e., no pairs occur). First, ignoring suits, the total number of possible sets (combinations) that form a straight with 7 distinct ranks is found. As with straight flushes, a straight is defined by its high card. With an ace-high straight, the ranks of the 2 extra cards may be chosen from any of the remaining 8 ranks, while with the 9 other possible straights, any of the ranks but the rank directly above the high card may be chosen. Thus, the total number of sets of ranks is:
Next, the total number of possible sets of suits, for any of the sets of ranks, is found. Given that each card is of a distinct rank, the total number of sets of suits is:
However, the instances where a flush is formed must be subtracted from the total; there are 3 ways of achieving this: There is 1 case per suit where all 7 are of the same suit. If 6 of the 7 are in the same suit, then the remaining card is chosen from the remaining 3 suits. If 5 of the 7 in the same suit, then 2 independent choices are made for each of the extra cards. Thus, the total number of cases where a flush is formed with 7 distinct ranks is:
Thus, the total number of sets of suits which produce a straight, but not a flush is:
And as each set of suits occurs for each set of ranks, the total number of straights with 7 distinct ranks is:
6 distinct ranks
A straight can also be formed with only 6 distinct ranks (i.e., the hand contains 1 pair). In this case, one of the extra cards will have the same rank as one of the cards forming the straight, therefore only one extra rank need be chosen. Thus, the total number of sets of ranks is:
The way to proceed now is to calculate the total number of ways to form a pair, and then calculate the total number of ways to form a straight, but not a flush (given that the pair has already been chosen). The pair can be 1 of the 6 previously chosen ranks, and 2 of the 4 of that rank form the pair. Thus, the total number of ways for form a pair is:
The total number of sets of suits for the remaining 5 cards can be calculated in the same way as for 7 cards:
As with 7 distinct ranks, the instances where a flush is formed must be subtracted from the total. The remaining 5 cards can be chosen in two different manners in order to form a flush: either they are all of the same suit, or 4 of them are in the same suit as either of the two paired cards. If all 5 are in the same suit, 1 of the 4 suits is chosen. If 4 of the 5 are in the same suit, 1 of the 2 suits forming the pair is chosen, and the suit of the extra card is chosen from the remaining 3 suits. Thus the total number of ways to form a flush is:
Thus, the total number of sets of suits which produce a straight, but not a flush is:
Thus the total number of straights with 6 distinct ranks equals the total number of sets of ranks, multiplied by the total number of ways to form the pair, multiplied by the total number of ways to form a straight:
5 distinct ranks with a triple
There are two ways to form a straight with 5 distinct ranks. The first is using 3 cards of the same rank, and 4 of separate ranks. There are only 10 sets of ranks in this case, as there are no extra ranks to be chosen. The triple can be 1 of the 5 ranks, and 3 of the 4 of that rank make up the triple. Thus, the number of ways to choose the triple is:
The total number of sets of suits for the remaining 4 cards is and the only ways to form a flush are if all 4 cards are of the same suit as 1 of the 3 suits forming the triple. Thus, the total number of straights form a straight, but not a flush is:
Thus the total number of straights with 5 distinct ranks and a triple is:
5 distinct ranks with 2 pairs
The second way to form a straight with 5 distinct ranks is to have 2 pairs and 3 other cards of separate ranks. As before, there are 10 different sets of ranks, however, calculating the number of ways that a flush is formed is complicated, due to the fact that the two pairs can consist of either 2, 3 or 4 suits. Firstly, the ranks for the two pairs are chosen from the 5 available. Thus, the number of ways to choose the ranks for the two pairs is:
Then the cards are chosen for each of the pairs. Thus, the number of ways to choose the suits for the pairs is:
6 of these ways, the pairs consist of 2 suits, 24 of these ways the pairs consist of 3 suits, and the remaining 6 of these ways they consist of 4 suits. Note that the total number of sets of suits for the remaining 3 cards is . When the pairs consist of 2 suits, a flush will be formed when the remaining 3 cards are all in either of those two suits. There are 2 ways of this happening which must be subtracted from the total. When there are 3 suits, a flush will be formed when the remaining 3 cards are all in the suit of the 2 cards of matching suit in the pairs. There is 1 way of this happening. When there are 4 suits there are no ways of making a flush. Thus, the total number of sets of suits that do not form a flush is:
Thus, the total number of straights with 5 distinct ranks and 2 pairs is:
Thus, the total number of straights is:
The rank of the triple is chosen from the 5 available, and 3 of the 4 of that suit are chosen. Thus, the total number of ways of choosing the triple is:
There are ways to choose the suits of the remaining 4 cards, minus the ways in which all 4 match one of the 3 suits in the triple (making a flush):
Thus, the total number of three of a kinds is:
3 pairs with 1 kicker
The 4 ranks are chosen, then 3 of the 4 are chosen for the 3 pairs, and 2 of the 4 of each rank are chosen for each pair. The kicker is then chosen from the 4 cards in the remaining rank. Thus, the total number of 3 pairs with 1 kicker is:
2 pairs with 3 kickers
A two pair hand must consist of 5 of the 13 ranks, but the 10 combinations that form straights must be subtracted. 2 of the ranks are chosen for the pairs and as with the calculations for straights with 5 ranks and two pairs, there are 2,268 sets of suits that do not form flushes. Thus, the total number of 2 pairs with 3 kickers is:
Thus, the total number of two pairs is:
There are ways of choosing the ranks of the kickers, and as with the calculations for straights with 6 distinct suits, there are 34 sets of suits that form flushes, therefore the total number of sets of suits that do not form flushes is:
There are 6 different ranks to choose for the pair and the pair can be formed from 2 of the 4 cards in that rank, therefore the number of ways to choose the pair is:
Thus, the total number of pair hands is:
There are ways of choosing the suits of the cards, and as with the calculations for straights with 7 distinct suits, there are 844 sets of suits that form flushes, therefore the total number of sets of suits that do not form flushes is:
Thus, the total number of no pair hands is:

Frequency of 5-card lowball poker hands

Some variants of poker, called lowball, use a low hand to determine the winning hand. 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 5-card combinations. (The frequencies given are exact; the probabilities and odds are approximate.)

Hand Distinct hands Frequency Probability Cumulative Odds
5-high 1 1,024 0.0394% 0.0394% 2,537.05 : 1
6-high 5 5,120 0.197% 0.236% 506.61 : 1
7-high 15 15,360 0.591% 0.827% 168.20 : 1
8-high 35 35,840 1.38% 2.21% 71.52 : 1
9-high 70 71,680 2.76% 4.96% 35.26 : 1
10-high 126 129,024 4.96% 9.93% 19.14 : 1
Jack-high 210 215,040 8.27% 18.2% 11.09 : 1
Queen-high 330 337,920 13.0% 31.2% 6.69 : 1
King-high 495 506,880 19.5% 50.7% 4.13 : 1
Total 1,287 1,317,888 50.7% 50.7% 0.97 : 1

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:

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

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:

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

and the frequency of low hands that are -high is .

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 combinations of ranks that yield a straight. This gives the following frequency for low hands of rank that do not include a straight or a flush:

Frequency of 7-card lowball poker hands

In some variants of poker a player uses the best five-card low hand selected 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 table does not extend to include five-card hands with at least one pair. Its "Total" represents 95.4% of the time that a player can select a 5-card low hand without any pair.

Hand Frequency Probability Cumulative Odds
5-high 781,824 0.584% 0.584% 170.12 : 1
6-high 3,151,360 2.36% 2.94% 41.45 : 1
7-high 7,426,560 5.55% 8.49% 17.01 : 1
8-high 13,171,200 9.85% 18.3% 9.16 : 1
9-high 19,174,400 14.3% 32.7% 5.98 : 1
10-high 23,675,904 17.7% 50.4% 4.65 : 1
Jack-high 24,837,120 18.6% 68.9% 4.39 : 1
Queen-high 21,457,920 16.0% 85.0% 5.23 : 1
King-high 13,939,200 10.4% 95.4% 8.60 : 1
Total 127,615,488 95.4% 95.4% 0.05 : 1

(The frequencies given are exact; the probabilities and odds are approximate.)

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 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:

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 , so this type of hand can only occur when there are at least two ranks greater than - 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:

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 , so this type of hand can only occur when there is at least one rank greater than - 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:

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:
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:

Thus there are ways to make a low hand with five distinct ranks.

Derivation

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

The total number of queen-high low hands is:

The total number of king-high low hands is:

Bluffing

Bluffing frequency can affect the probability of an opponent calling a bet or folding. A player can bluff at an § Optimal bluffing frequency to try to eliminate any advantage to their opponent.

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