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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 鈿撅笍 the proportion of hands of that type among all
possible hands.
History [ edit ]
Probability and gambling have been ideas since 鈿撅笍 long
before the invention of poker. The development of probability theory in the late 1400s
was attributed to gambling; when 鈿撅笍 playing a game with high stakes, players wanted to
know what the chance of winning would be. In 1494, Fra 鈿撅笍 Luca Paccioli released his work
Summa de arithmetica, geometria, proportioni e proportionalita which was the first
written text on probability. 鈿撅笍 Motivated by Paccioli's work, Girolamo Cardano (1501-1576)
made further developments in probability theory. His work from 1550, titled Liber de
鈿撅笍 Ludo Aleae, discussed the concepts of probability and how they were directly related to
gambling. However, his work did not 鈿撅笍 receive any immediate recognition since it was not
published until after his death. Blaise Pascal (1623-1662) also contributed to
probability 鈿撅笍 theory. His friend, Chevalier de M茅r茅, was an avid gambler with the goal to
become wealthy from it. De M茅r茅 鈿撅笍 tried a new mathematical approach to a gambling game
but did not get the desired results. Determined to know why 鈿撅笍 his strategy was
unsuccessful, he consulted with Pascal. Pascal's work on this problem began an
important correspondence between him and 鈿撅笍 fellow mathematician Pierre de Fermat
(1601-1665). Communicating through letters, the two continued to exchange their ideas
and thoughts. These interactions 鈿撅笍 led to the conception of basic probability theory. To
this day, many gamblers still rely on the basic concepts of 鈿撅笍 probability theory in order
to make informed decisions while gambling.[1][2]
Frequencies [ edit ]
5-card poker
hands [ edit ]
An Euler diagram 鈿撅笍 depicting poker hands and their odds from a typical
American 9/6 Jacks or Better machine
In straight poker and five-card draw, 鈿撅笍 where there
are no hole cards, players are simply dealt five cards from a deck of 52.
The following
chart enumerates 鈿撅笍 the (absolute) frequency of each hand, given all combinations of five
cards randomly drawn from a full deck of 52 鈿撅笍 without replacement. Wild cards are not
considered. In this chart:
Distinct hands is the number of different ways to draw the
鈿撅笍 hand, not counting different suits.
is the number of different ways to draw the hand,
not counting different suits. Frequency is 鈿撅笍 the number of ways to draw the hand,
including the same card values in different suits.
is the number of ways 鈿撅笍 to draw the
hand, the same card values in different suits. The Probability of drawing a given hand
is calculated 鈿撅笍 by dividing the number of ways of drawing the hand ( Frequency ) by the
total number of 5-card hands 鈿撅笍 (the sample space; ( 52 5 ) = 2 , 598 , 960 {\textstyle
{52 \choose 5}=2,598,960} 4 / 2,598,960 鈿撅笍 , or one in 649,740. One would then expect to
draw this hand about once in every 649,740 draws, or 鈿撅笍 nearly 0.000154% of the time.
of
drawing a given hand is calculated by dividing the number of ways of drawing the 鈿撅笍 hand (
) by the total number of 5-card hands (the sample space; , or one in 649,740. One would
鈿撅笍 then expect to draw this hand about once in every 649,740 draws, or nearly 0.000154% of
the time. Cumulative probability 鈿撅笍 refers to the probability of drawing a hand as good as
or better than the specified one. For example, the 鈿撅笍 probability of drawing three of a
kind is approximately 2.11%, while the probability of drawing a hand at least as 鈿撅笍 good
as three of a kind is about 2.87%. The cumulative probability is determined by adding
one hand's probability with 鈿撅笍 the probabilities of all hands above it.
refers to the
probability of drawing a hand as good as the specified one. 鈿撅笍 For example, the
probability of drawing three of a kind is approximately 2.11%, while the probability of
drawing a hand 鈿撅笍 as good as three of a kind is about 2.87%. The cumulative probability is
determined by adding one hand's probability 鈿撅笍 with the probabilities of all hands above
it. The Odds are defined as the ratio of the number of ways 鈿撅笍 not to draw the hand, to
the number of ways to draw it. In statistics, this is called odds against 鈿撅笍 . For
instance, with a royal flush, there are 4 ways to draw one, and 2,598,956 ways to draw
something 鈿撅笍 else, so the odds against drawing a royal flush are 2,598,956 : 4, or 649,739
: 1. The formula for 鈿撅笍 establishing the odds can also be stated as (1/p) - 1 : 1 , where
p is the aforementioned probability.
are 鈿撅笍 defined as the ratio of the number of ways to
draw the hand, to the number of ways to draw 鈿撅笍 it. In statistics, this is called . For
instance, with a royal flush, there are 4 ways to draw one, 鈿撅笍 and 2,598,956 ways to draw
something else, so the odds against drawing a royal flush are 2,598,956 : 4, or 鈿撅笍 649,739
: 1. The formula for establishing the odds can also be stated as , where is the
aforementioned probability. 鈿撅笍 The values given for Probability, Cumulative probability,
and Odds are rounded off for simplicity; the Distinct hands and Frequency values 鈿撅笍 are
exact.
The nCr function on most scientific calculators can be used to calculate hand
frequencies; entering nCr with 52 and 鈿撅笍 5 , for example, yields ( 52 5 ) = 2 , 598 , 960
{\textstyle {52 \choose 5}=2,598,960} as 鈿撅笍 above.
The royal flush is a case of the
straight flush. It can be formed 4 ways (one for each suit), 鈿撅笍 giving it a probability of
0.000154% and odds of 649,739 : 1.
When ace-low straights and ace-low straight flushes
are not 鈿撅笍 counted, the probabilities of each are reduced: straights and straight flushes
each become 9/10 as common as they otherwise would 鈿撅笍 be. The 4 missed straight flushes
become flushes and the 1,020 missed straights become no pair.
Note that since suits
have 鈿撅笍 no relative value in poker, two hands can be considered identical if one hand can
be transformed into the other 鈿撅笍 by swapping suits. For example, the hand 3鈾?7鈾?8鈾?Q鈾?A鈾?/p>
is identical to 3鈾?7鈾?8鈾?Q鈾?A鈾?鈿撅笍 because replacing all of the clubs in the first hand
with diamonds and all of the spades with hearts produces 鈿撅笍 the second hand. So
eliminating identical hands that ignore relative suit values, there are only 134,459
distinct hands.
The number of 鈿撅笍 distinct poker hands is even smaller. For example, 3鈾?7鈾?/p>
8鈾?Q鈾?A鈾?and 3鈾?7鈾?8鈾?Q鈾?A鈾?are 鈿撅笍 not identical hands when just ignoring suit assignments
because one hand has three suits, while the other hand has only 鈿撅笍 two鈥攖hat difference
could affect the relative value of each hand when there are more cards to come.
However, even though 鈿撅笍 the hands are not identical from that perspective, they still form
equivalent poker hands because each hand is an A-Q-8-7-3 鈿撅笍 high card hand. There are
7,462 distinct poker hands.
7-card poker hands [ edit ]
In some popular variations of
poker such 鈿撅笍 as Texas hold 'em, the most widespread poker variant overall,[3] a player
uses the best five-card poker hand out of 鈿撅笍 seven cards.
The frequencies are calculated
in a manner similar to that shown for 5-card hands,[4] except additional complications
arise due 鈿撅笍 to the extra two cards in the 7-card poker hand. The total number of distinct
7-card hands is ( 52 鈿撅笍 7 ) = 133,784,560 {\textstyle {52 \choose 7}=133{,}784{,}560} . It
is notable that the probability of a no-pair hand is 鈿撅笍 lower than the probability of a
one-pair or two-pair hand.
The Ace-high straight flush or royal flush is slightly more
frequent 鈿撅笍 (4324) than the lower straight flushes (4140 each) because the remaining two
cards can have any value; a King-high straight 鈿撅笍 flush, for example, cannot have the Ace
of its suit in the hand (as that would make it ace-high instead).
(The 鈿撅笍 frequencies
given are exact; the probabilities and odds are approximate.)
Since suits have no
relative value in poker, two hands can 鈿撅笍 be considered identical if one hand can be
transformed into the other by swapping suits. Eliminating identical hands that ignore
鈿撅笍 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 鈿撅笍 is 4,824. Perhaps surprisingly, this
is fewer than the number of 5-card poker hands from 5 cards, as some 5-card 鈿撅笍 hands are
impossible with 7 cards (e.g. 7-high and 8-high).
5-card lowball poker hands [ edit
]
Some variants of poker, called 鈿撅笍 lowball, use a low hand to determine the winning hand.
In most variants of lowball, the ace is counted as 鈿撅笍 the lowest card and straights and
flushes don't count against a low hand, so the lowest hand is the five-high 鈿撅笍 hand
A-2-3-4-5, also called a wheel. The probability is calculated based on ( 52 5 ) = 2 ,
598 鈿撅笍 , 960 {\textstyle {52 \choose 5}=2,598,960} , the total number of 5-card
combinations. (The frequencies given are exact; the probabilities 鈿撅笍 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 鈿撅笍 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% 鈿撅笍 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 鈿撅笍 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% 鈿撅笍 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 鈿撅笍 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 鈿撅笍 rotate the hand descriptions so
that 6-high replaces 5-high for the best hand and ace-high replaces king-high as the
worst 鈿撅笍 hand.
Some players do not ignore straights and flushes when computing the low
hand in lowball. In this case, the lowest 鈿撅笍 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 鈿撅笍 listed",
"6-high" has one distinct hand, and "King-high" having 330 distinct hands,
respectively. The Total line also needs adjusting.
7-card lowball 鈿撅笍 poker hands [ edit
]
In some variants of poker a player uses the best five-card low hand selected from
seven 鈿撅笍 cards. In most variants of lowball, the ace is counted as the lowest card and
straights and flushes don't count 鈿撅笍 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 鈿撅笍 calculated based on (
52 7 ) = 133 , 784 , 560 {\textstyle {52 \choose 7}=133,784,560} , the total 鈿撅笍 number of
7-card combinations.
The table does not extend to include five-card hands with at least
one pair. Its "Total" represents 鈿撅笍 the 95.4% of the time that a player can select a
5-card low hand without any pair.
Hand Frequency Probability Cumulative 鈿撅笍 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% 鈿撅笍 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% 鈿撅笍 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 鈿撅笍 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 鈿撅笍 are approximate.)
If aces are not low,
simply rotate the hand descriptions so that 6-high replaces 5-high for the best hand
鈿撅笍 and ace-high replaces king-high as the worst hand.
Some players do not ignore straights
and flushes when computing the low hand 鈿撅笍 in lowball. In this case, the lowest hand is
A-2-3-4-6 with at least two suits. Probabilities are adjusted in the 鈿撅笍 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. 鈿撅笍 The Total line also needs adjusting.
See also
[ edit ]
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