What is the theoretical probability of being dealt all four 7s in a 5 card hand
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Probability of getting 4 aces out of a hand of 5
If you have a standard deck of 52 cards, what is the probability that out of a hand of 5 cards you get 4 aces? First I found the total # of ways for choosing 5 cards from 52 = (52 C 5) = 2,598,960 Did I do this right? Answers and Replies
Suggested for: Probability of getting 4 aces out of a hand of 5
(most recent edit: January 2, 2005)A SINGLE PAIRThis 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 PAIRThis 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 TRIPLEThis 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 HOUSEThis 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 KINDThis 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 STRAIGHTThis 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 0.00392465A FLUSHHere 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 FLUSHAll 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 FLUSHThis 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 ABOVEWe 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.
What is the probability of getting 4 of a kind in a 5 card hand?FOUR OF A KIND
The number of such hands is (13-choose-1)(4-choose-4)(12-choose-1)(4-choose-1). The probability is 0.000240.
What is a 5 card hand probability?There are 2,598,960 many possible 5-card Poker hands. Thus the probability of obtaining any one specific hand is 1 in 2,598,960 (roughly 1 in 2.6 million).
How many ways can a 5 card hand be dealt?First, count the number of five-card hands that can be dealt from a standard deck of 52 cards. We did this in the previous section, and found that there are 2,598,960 distinct poker hands.
How many 5 card poker hands contain a 4 of a kind?of ranks, there are 4 choices for each card except we cannot choose all in the same suit. Hence, there are 1277(45-4) = 1,302,540 high card hands.
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