 
5CARD POKER HANDS
(http://www.math.hawaii.edu/~ramsey/Probability/PokerHands.html)
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
(13choose1)*(4choose2)*(12choose3)*[(4choose1)]^3.
If all hands are equally likely, the probability of a single pair is
obtained by dividing by (52choose5). 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
(13choose2)(4choose2)(4choose2)(11choose1)(4choose1).
After dividing by (52choose5), 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
(13choose1)(4choose3)(12choose2)[4choose1]^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
(13choose1)(4choose3)(12choose1)(4choose2). 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
(13choose1)(4choose4)(12choose1)(4choose1). 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*[4choose1]^5.
The probability is 0.003940.
A FLUSH Here all 5 cards are from the same suit
(they may also be a straight). The number of such hands is
(4choose1)*
(13choose5). The probability is 0.001981.
A STRAIGHT FLUSH All 5 cards are from the same suit
and they form a straight. The number of such hands is 4*10, and the
probability is 0.000015.
A ROYAL FLUSH This consists of the ten, jack,
queen,
king, and ace of one suit. There are four such hands. The
probability
is 0.000002.
NONE OF THE ABOVE We have to choose 5 distinct
kinds
(13choose5) 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: 4^54. The total number of such hands is
[(13choose5)10]*
(4^54). The probability is 0.501177.
