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?
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 matching 2 of a kind on the board: (48 choose 3)/(50 choose 5)
Probability of no pocket pair: 1248/1326
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 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
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.
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.
