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Several years ago, I was playing Limit Hold'Em at the Palms. I had pocket sixes, and ended up turning those into quads. Then, the very next hand -- from a different, auto-shuffled deck -- I got pocket sixes again, which again turned into quads! I got the same four-of-a-kind two hands in a row. I figure the odds against that are astronomical.

And I want to know, just how astronomical are they? Just what exactly are the odds of getting the same four-of-a-kind twice in a row in Hold'em?

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up vote 6 down vote accepted

The odds of getting a 4 of a kind given 7 cards (2 in your hand and 5 on the board) are (13 * (48 choose 3)) / (52 choose 7) or 0.00168067227. The probability of getting that specific 4 of a kind again are now (48 choose 3) / (52 choose 7) or 0.000129282482.

The probability of both events occurring is 0.00168067227* 0.000129282482, which is 0.000000217281482.

Note that this leaves open the possibility that other people at the table would also still have the four of a kind (as in, it includes those situations where there is a four of a kind on the board.) Presumably, you'd want to exclude those.

For this, we have to multiply the probability of getting 3 of a kind on the board while having an unpaired hand with the probability of getting 2 of a kind on the board while having a paired hand.

  • Probability of a pocket pair = 78/1326
  • Probability of matching 2 of a kind on the board: (48 choose 3)/(50 choose 5)

  • Probability of no pocket pair: 1248/1326

  • Probability of matching 3 of a kind on the board: (47 choose 2)/(50 choose 5)

So the probability of an exclusive four of a kind is:

(48 choose 3)/(50 choose 5) * (78/1326) + (47 choose 2)/(50 choose 5) * (1248/1326) = 0.000960384154

The probability of getting the same exact four of a kind is then:

  • Probability of same pocket pair = 6/1326
  • Probability of matching 2 of a kind on the board: (48 choose 3)/(50 choose 5)

  • Probability of no pocket pair that matches that 4 of a kind: 208/1326

  • Probability of matching 3 of a kind on the board: (47 choose 2)/(50 choose 5)

This is:

(48 choose 3)/(50 choose 5) * (6/1326) + (47 choose 2)/(50 choose 5) * (208/1326) = 0.000116969865.

So the final probability is 0.000116969865 * 0.000960384154 = 0.000000112336005.

So basically: 8,901,864 : 1 odds of you getting the same four of a kind assuming you are the only one with the four of a kind each time. Otherwise, 4,602,324 : 1 odds of you getting the same four of a kind twice.

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Thank you! This explains it really well. So, if I wanted to limit it to four-of-a-kind starting with a pocket pair for both hands, it would be the same process, except not adding in the no-pocket-pair odds, right? I think if I'm doing my math right, the odds against this were 56,378,480:1 ... or to put it another way, I was more likely to win the California Super Lotto jackpot than I was to get this sequence of hands. –  SkippyKawakami Jan 27 '12 at 0:41
    
@SkippyKawakami: If only you could play the lottery as often as you play a poker hand... You'd be able to lose a whole lot of money! –  configurator Jan 10 '13 at 19:56
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Hitting quads is about 122 to 1 by the river with a pocket pair. Without going through the math exactly, it would be into the 1,000,000+ to 1 region. That may be broad but your question is also! Nice to have you here @SkippyKawakami. Please see the FAQ:What not to ask about more appropriate questions for members to get stuck into.

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Just out of curiosity, why is my question broad? I thought it would have a precise answer, and a precise method of determining that answer. But if I was ambiguous, I'm happy to edit the question for clarity. –  SkippyKawakami Jan 24 '12 at 23:46
    
You are right to say that this particular question about quads has a precise answer. I was really alluding to the conceptual broadness of the question. You could equally ask the same thing and substitute "four-of-a-kind" with any other combination of hand types, like two-pair, sets, etc. The conceptual scope of an odds question like this covers so much, and yet your specific question covers so little to be of any use to many others. That is unless it helps to plan on hitting back-to-back quads regularly (and I do!). –  Toby Booth Jan 25 '12 at 0:07
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