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.
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 ]
