(most current edit: January 2, 2005)
A single PAIRThis the hand through the pattern AABCD,where A, B, C and also D are from the distinct "kinds" of cards: aces,twos, threes, tens, jacks, queens, and kings (there space 13 kinds,and 4 of each kind, in the conventional 52 map deck). The number ofsuch hands is (13-choose-1)*(4-choose-2)*(12-choose-3)*<(4-choose-1)>^3.If every hands are equally likely, the probability the a solitary pair isobtained by splitting by (52-choose-5). This probability is 0.422569.
You are watching: What is the probability that a five-card poker hand contains
TWO PAIRThis hand has actually the pattern AABBC where A, B,and C room from distinctive kinds. The number of such hand is(13-choose-2)(4-choose-2)(4-choose-2)(11-choose-1)(4-choose-1).After splitting by (52-choose-5), the probability is 0.047539.
A TRIPLEThis hand has the pattern AAABC wherein A, B,and C are from distinctive kinds. The number of such hands is(13-choose-1)(4-choose-3)(12-choose-2)<4-choose-1>^2. The probabilityis 0.021128.
See more: Exactly How Many Cups In A Pound Of Potato Salad To Buy For Any Group Size
A full HOUSEThis hand has the pattern AAABB whereA and also B are from unique kinds. The variety of such hands is(13-choose-1)(4-choose-3)(12-choose-1)(4-choose-2). The probabilityis 0.001441.
four OF A typeThis hand has the pattern AAAAB whereA and B room from distinctive kinds. The variety of such hands is(13-choose-1)(4-choose-4)(12-choose-1)(4-choose-1). The probabilityis 0.000240.
A directlyThis is five cards in a sequence (e.g.,4,5,6,7,8), with aces enabled to be either 1 or 13 (low or high) andwith the cards allowed to it is in of the very same suit (e.g., all hearts) orfrom some various suits. The number of such hand is 10*<4-choose-1>^5.The probability is 0.003940. IF YOU average TO EXCLUDE straight FLUSHESAND royal FLUSHES (SEE BELOW), the variety of such hands is 10*<4-choose-1>^5 - 36 - 4 = 10200, with probability 0.00392465
A flushright here all 5 cards room from the exact same suit(they may likewise be a straight). The variety of such hands is (4-choose-1)*(13-choose-5). The probability is roughly 0.00198079. IF YOU median TO EXCLUDE straight FLUSHES, SUBTRACT 4*10 (SEE THE following TYPEOF HAND): the variety of hands would then it is in (4-choose-1)*(13-choose-5)-4*10,with probability around 0.0019654.
A directly FLUSHevery 5 cards room from the same suitand they form a right (they may also be a imperial flush). The number of such hands is 4*10, and theprobability is 0.0000153908. IF YOU median TO EXCLUDE imperial FLUSHES, SUBTRACT 4(SEE THE NEXT type OF HAND): the number of hands would then be 4*10-4 = 36, with probability approximately0.0000138517.
A imperial FLUSHThis consists of the ten, jack, queen,king, and also ace of one suit. Over there are 4 such hands. The probabilityis 0.00000153908.