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5-CARD POKER HANDS
(most recent edit: January 2, 2005)
A SINGLE PAIR This the hand with the pattern AABCD,
where A, B, C and D are from the distinct "kinds" of cards: aces,
twos, threes, tens, jacks, queens, and kings (there are 13 kinds,
and four of each kind, in the standard 52 card deck). The number of
such hands is (13-choose-1)*(4-choose-2)*(12-choose-3)*[(4-choose-1)]^3.
If all hands are equally likely, the probability of a single pair is
obtained by dividing by (52-choose-5). This probability is 0.422569.
TWO PAIR This hand has the pattern AABBC where A, B,
and C are from distinct kinds. The number of such hands is
(13-choose-2)(4-choose-2)(4-choose-2)(11-choose-1)(4-choose-1).
After dividing by (52-choose-5), the probability is 0.047539.
A TRIPLE This hand has the pattern AAABC where A, B,
and C are from distinct kinds. The number of such hands is
(13-choose-1)(4-choose-3)(12-choose-2)[4-choose-1]^2. The probability
is 0.021128.
A FULL HOUSE This hand has the pattern AAABB where
A and B are from distinct kinds. The number of such hands is
(13-choose-1)(4-choose-3)(12-choose-1)(4-choose-2). The probability
is 0.001441.
FOUR OF A KIND This hand has the pattern AAAAB where
A and B are from distinct kinds. The number of such hands is
(13-choose-1)(4-choose-4)(12-choose-1)(4-choose-1). The probability
is 0.000240.
A STRAIGHT This is five cards in a sequence (e.g.,
4,5,6,7,8), with aces allowed to be either 1 or 13 (low or high) and
with the cards allowed to be of the same suit (e.g., all hearts) or
from some different suits. The number of such hands is 10*[4-choose-1]^5.
The probability is 0.003940. IF YOU MEAN TO EXCLUDE STRAIGHT FLUSHES
AND ROYAL FLUSHES (SEE BELOW), the number of such hands is 10*[4-choose-1]^5
- 36 - 4 = 10200, with probability 0.00392465
A FLUSH Here all 5 cards are from the same suit
(they may also be a straight). The number of such hands is (4-choose-1)*
(13-choose-5). The probability is approximately 0.00198079.
IF YOU MEAN TO EXCLUDE STRAIGHT FLUSHES, SUBTRACT 4*10 (SEE THE NEXT TYPE
OF HAND): the number of hands would then be (4-choose-1)*(13-choose-5)-4*10,
with probability approximately 0.0019654.
A STRAIGHT FLUSH All 5 cards are from the same suit
and they form a straight (they may also be a royal flush). The number of such hands is 4*10, and the
probability is 0.0000153908. IF YOU MEAN TO EXCLUDE ROYAL FLUSHES, SUBTRACT 4
(SEE THE NEXT TYPE OF HAND): the number of hands would then be 4*10-4 = 36, with probability approximately
0.0000138517.
A ROYAL FLUSH This consists of the ten, jack, queen,
king, and ace of one suit. There are four such hands. The probability
is 0.00000153908.
NONE OF THE ABOVE We have to choose 5 distinct kinds
(13-choose-5) but exclude any straights (subtract 10). We can have any
pattern of suits except the 4 patterns where all 5 cards have the
same suit: 4^5-4. The total number of such hands is [(13-choose-5)-10]*
(4^5-4). The probability is 0.501177.
| Hand | Probability | Number of Hands |
| Single Pair | 0.422569 | 1098240 |
| Two Pair | 0.047539 | 123552 |
| Triple | 0.0211285 | 54912 |
| Full House | 0.00144058 | 3744 |
| Four of a Kind | 0.000240096 | 624 |
Straight
(excluding Straight Flush
and Royal Flush) | 0.00392465 | 10200 |
| Flush (but not a Straight) | 0.0019654 | 5108 |
| Straight Flush (but not Royal) | 0.0000138517 | 36 |
| Royal Flush | 0.00000153908 | 4 |
| None of the Above | 0.501177 | 1302540 |
| Sum over except this list
| 0.999999616
| 2598960 |
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