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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 M�r�, was an avid gambler with the goal to

become wealthy from it. De M�r� 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.

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; , 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.

See also

[ edit ]