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