A Poker Player reader asked me the odds against having quads twice in a row at Hold'Em, because he did so last month. He had been playing poker for 60 years, he wrote, and it hadn't happened before, so he guessed that it was rare. How rare was it really, he asked. Although Texas Hold'Em is more recent than that, it's still a good question.
There are two ways to make Four of a Kind: you can start with a pocket pair, and then have the other two appear among the community cards; you can start with an unpaired hand, and then have three of either rank appear on the tableau.
Given that you started with a pocket pair, the probability that the other two will appear on the table is C(2,2)*C(48,3)/C(50,5), which equals .00816, or about 1/122.5. The probability of having been dealt a pocket pair is 1/17. Before you peek at your hole cards, the probability that you'll have Four of a Kind this way is the product, .00048, or odds of about 2081 to 1 against.
With an unpaired start, a probability of 16/17, you'll make quads if three cards of either rank appear on the tableau after all the cards are out. That probability is given by 2*C(3,3)*C(47,2)/C(50,5), .00102, or odds of 979 to 1 against. Before you peek at your cards, the probability of Four of a Kind is the product of those probabilities, .00096, odds of about 1040 to 1 against.
You're twice as likely to make quads of the second kind than the first after all the community cards are out. That surprised me. I thought quads of the first kind were more likely, and they are, on the Flop. (The probability of flopping quads of the first kind equals 1/17*C(2,2)*C(48,1)/C(50,3), odds of about 6941 to 1 against. The probability of flopping quads of the second kind equals 16/17*2*C(3,3)/C(50,3), about 10,412 to 1 against.)
At poker, after the cards have been ordered randomly, the probability of getting quads right after you've just had quads is either 2081 or 1040 to 1 against, because the cards have no memory. So your chance of getting quads on the next hand remains the same whether you've just had quads or not. That's one way to look at it. Here are other ways.
If, before looking at your hole cards, you were to wager anyone that you would have successive quads of the first kind, then odds of 4,336,805 to 1 would be fair.
If, before looking at your hole cards, you were to wager anyone that you would have successive quads of the first kind and then the second kind, or vice versa, then odds of 2,168,402 to 1 would be fair. If, before looking at your hole cards, you were to wager anyone that you would have successive quads of the second kind, then odds of 1,084,201 to 1 would be fair.
Looking at it another way it's a 50-50 chance of getting successive quads: once in 3,006,045 hands for the first kind; once in 1,503,022 hands of the first kind and the second kind, or vice versa; and, once in 751,511 hands for the second kind.
A player would pick up 4,200,000 starting hands in 60 years, playing 2000 hours per year at 35 hands per hour at Hold'Em. Therefore, his chances of having successive quads even of the first kind would be more than 50-50.
