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Chances of card combinations in poker

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 [ edit ]

Probability and gambling have been ideas since 💸 long

before the invention of poker. The development of probability theory in the late 1400s

was attributed to gambling; 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 were directly related to

gambling. However, his work did not 💸 receive any immediate recognition since 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.[1][2]

Frequencies [ edit ]

5-card poker

hands [ edit ]

An Euler diagram 💸 depicting poker hands and their odds from a typical

American 9/6 Jacks or Better machine

In straight poker and five-card draw, 💸 where there

are no hole cards, players are simply dealt five cards from a deck of 52.

The following

chart enumerates 💸 the (absolute) frequency of each hand, given all combinations of five

cards randomly drawn from a full deck of 52 💸 without replacement. Wild cards are not

considered. In this chart:

Distinct hands is the number of different ways to draw the

💸 hand, not counting different suits.

is the number of different ways to draw the hand,

not counting different suits. Frequency is 💸 the number of ways to draw the hand,

including the same card values in different suits.

is the number of ways 💸 to draw the

hand, the same card values in different suits. The Probability of drawing a given hand

is calculated 💸 by dividing the number of ways of drawing the hand ( Frequency ) by the

total number of 5-card hands 💸 (the sample space; ( 52 5 ) = 2 , 598 , 960 {\textstyle

{52 \choose 5}=2,598,960} 4 / 2,598,960 💸 , or one in 649,740. One would then expect to

draw this hand about once in every 649,740 draws, or 💸 nearly 0.000154% of the time.

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; , or one in 649,740. One would

💸 then expect to draw this hand about once in every 649,740 draws, or nearly 0.000154% of

the time. Cumulative probability 💸 refers to the probability of drawing a hand as good as

or better than the specified one. For example, the 💸 probability of drawing three of a

kind is approximately 2.11%, while the probability of drawing a hand at least as 💸 good

as three of a kind is about 2.87%. The cumulative probability is determined by adding

one hand's probability with 💸 the probabilities of all hands above it.

refers to the

probability of drawing a hand as good as the specified one. 💸 For example, the

probability of drawing three of a kind is approximately 2.11%, while the probability of

drawing a hand 💸 as good as three of a kind is about 2.87%. The cumulative probability is

determined by adding one hand's probability 💸 with the probabilities of all hands above

it. The Odds are defined as the ratio of the number of ways 💸 not to draw the hand, to

the number of ways to draw it. In statistics, this is called odds against 💸 . For

instance, with a royal flush, there are 4 ways to draw one, and 2,598,956 ways to draw

something 💸 else, so the odds against drawing a royal flush are 2,598,956 : 4, or 649,739

: 1. The formula for 💸 establishing the odds can also be stated as (1/p) - 1 : 1 , where

p is the aforementioned probability.

are 💸 defined as the ratio of the number of ways to

draw the hand, to the number of ways to draw 💸 it. In statistics, this is called . For

instance, with a royal flush, there are 4 ways to draw one, 💸 and 2,598,956 ways to draw

something else, so the odds against drawing a royal flush are 2,598,956 : 4, or 💸 649,739

: 1. The formula for establishing the odds can also be stated as , where is the

aforementioned probability. 💸 The values given for Probability, Cumulative probability,

and Odds are rounded off for simplicity; the Distinct hands and Frequency values 💸 are

exact.

The nCr function on most scientific calculators can be used to calculate hand

frequencies; entering nCr with 52 and 💸 5 , for example, yields ( 52 5 ) = 2 , 598 , 960

{\textstyle {52 \choose 5}=2,598,960} as 💸 above.

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.

7-card poker hands [ edit ]

In some popular variations of

poker such 💸 as Texas hold 'em, the most widespread poker variant overall,[3] 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,[4] 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 ( 52 💸 7 ) = 133,784,560 {\textstyle {52 \choose 7}=133{,}784{,}560} . It

is notable that the probability of a no-pair hand is 💸 lower 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).

(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, as some 5-card 💸 hands are

impossible with 7 cards (e.g. 7-high and 8-high).

5-card lowball poker hands [ edit

]

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 ( 52 5 ) = 2 ,

598 💸 , 960 {\textstyle {52 \choose 5}=2,598,960} , 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 against 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, threes- or

fours-of-a-kind. (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.

Some players do not ignore straights and flushes when computing the low

hand in lowball. In this case, the lowest 💸 hand is A-2-3-4-6 with at least two suits.

Probabilities are adjusted in the above table such that "5-high" is not 💸 listed",

"6-high" has one distinct hand, and "King-high" having 330 distinct hands,

respectively. The Total line also needs adjusting.

7-card lowball 💸 poker hands [ edit

]

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 (

52 7 ) = 133 , 784 , 560 {\textstyle {52 \choose 7}=133,784,560} , 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 💸 the 95.4% of the time that a player can select a

5-card low hand without any pair.

Hand Frequency Probability Cumulative 💸 Odds against

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.

Some players do not ignore straights

and flushes when computing the low hand 💸 in lowball. In this case, the lowest hand is

A-2-3-4-6 with at least two suits. Probabilities are adjusted in the 💸 above table such

that "5-high" is not listed, "6-high" has 781,824 distinct hands, and "King-high" has

21,457,920 distinct hands, respectively. 💸 The Total line also needs adjusting.