(52)(51)(50)(49)(48) C552 = -------------------- = 2 598 960 hands (1)(2)(3)(4)(5)
♣ ♦ ♥ ♦ ♥ ♠ ♣ ♥ ♠ ♣ ♦ ♠
There are 13 denominations resulting in 4 x 13 = 52 ways to get three cards of one denomination. The pair must be in another denomination. There are 12 possible denominations, and 6 ways of pairing the denominations:
♣ ♦ ♣ ♥ ♣ ♠ ♦ ♥ ♦ ♠ ♥ ♠
Thus, the pair can be obtained in 6 x 12 = 72 ways. In total, there are 52 x 72 = 3 744 ways of making a Full House
(13)(12)(11)(10)(9) 4*C513 = 4 * ------------------- = 5 148 combinations (1)(2)(3)(4)(5)
C24 x C24 x C213 x 44 = 6 x 6 x 13 x 12 x 44/2 = 123 552 combinations
13 x C24 x 48 x 44 x 40 / (3 x 2 x 1) = 1 098 240 combinations
Dividing each number by the total number of hands (2 598 960) yields the probabilities for each poker hand for the initial deal:
----------------------------------------------- Hand Probability % ----------------------------------------------- Royal Flush 0.000002 0.0002 Straight Flush 0.000014 0.0014 Four of a Kind 0.00024 0.0240 Full House 0.00144 0.144 Flush 0.00197 0.197 Straight 0.00392 0.392 Three of a Kind 0.02113 2.113 Two Pairs 0.04754 4.754 One Pair 0.42257 42.257 High Card 0.50112 50.112 -----------------------------------------------
Poker is a game that is German in origin, and gets its name from the German pochen, meaning "to boast or brag"- though the literal translation is "to knock." A "knock" is still used in poker to indicate a passed bet, and the game still includes considerable "bragging." However, the game appears to have been introduced to this country through New Orleans, where, due to its similarity to the French card game of poque, it acquired that name. Southern gentlemen who played the game but did not know French pronunciation tried to say "pok-uh", which is very similar to the pronunciation of "poker" today. Northerners that learned the game deduced, therefore, that it was spelled like it sounds.
Pok"er (?), n. [From Poke to push.]
1.
One who pokes.
2.
That which pokes or is used in poking, especially a metal bar or rod used in stirring a fire of coals.
3.
A poking-stick.
Decker.
4. Zool.
The poachard.
Poker picture, a picture formed in imitation of bisterwashed drawings, by singeing the surface of wood with a heated poker or other iron.
Fairholt.
© Webster 1913.
Pok"er, n. [Of uncertain etymol.]
A game at cards derived from brag, and first played about 1835 in the Southwestern United States.
Johnson's Cyc.
Pok"er, n. [Cf. Dan. pokker the deuce, devil, also W. pwci, a hobgoblin, bugbear, and E. puck.]
Any imagined frightful object, especially one supposed to haunt the darkness; a bugbear.
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