Poker, Probability, CombinatoricsDate: 11/04/97 at 14:40:51 From: Hein Hundal Subject: Poker, Probability, Combinatorics Recently on rec.gambling.poker, we were discussing a question that was simple to pose, but fairly hard to answer. Question: If we deal n hands consisting of 2 cards each, what is the probability that there will be no pairs amoung the hands? For example: If hand1 = { Jack of clubs, Queen of clubs }, and hand2 = { Jack of spades, 3 of diamonds}, then two hands were dealt and no pairs appeared. One poster figured out an algorithm the gives the answers, but I was hoping there might be a power theorem that gives the probability in terms of the number of values and suits in the card deck. For a standard card deck with values 2,3,...,9,10, Jack, Queen, King, Ace and 4 suits, the probabilities were 16 18448 48008 88348248 715247168 {--, -----, -----, ---------, ---------} 17 20825 57575 112559125 968008475 for 1,2,3,4, and 5 hands respectively. Hein Hundal Date: 11/04/97 at 18:11:45 From: Doctor Anthony Subject: Re: Poker, Probability, Combinatorics With just one pair, we can choose the first card in 52 ways, and the second card in 48 ways. So the probability of no pair is 52 x 48 48 16 ------- = ---- = --- 52 x 51 51 17 For 2 hands, the probability that the first hand is not a pair is 16/17. For the second pair there are several possibilities depending on whether the face values are completely different from those in the first hand, or whether they repeat some of those values. If completely different, the number of ways of selecting the hand = 44 x 40. If one card is a repeat, the number of ways = 6 x 44 + 44 x 6 If two cards are repeats, the number of ways = 6 x 3 16[44 x 40 + 12 x 44 + 6 x 3] 36896 Total probability = ----------------------------- = --------- 17 x 50 x 49 41650 18448 = ------ 20825 As you can see, the calculation is fairly laborious, as the various possibilities have to be considered. There is no simple formula for n hands. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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