Quads on the turn

At the final table of the Mookie tonight we witnessed this unusual hand. This screen shot makes it look like I was involved, though I wasn't. I folded pre-flop. I don't remember who else played it, but it was won by a bet on the turn without a showdown.

It made me wonder how rare it is to see four of a kind in the first four board cards. Let's find out, shall we?

As I have discussed before, there are 19,600 possible flops, given that I am holding two cards that are therefore unavailable. How many of these flops will be three of a kind? Well, there are 11 ranks of cards not matching the ones that I'm holding (assuming I don't have a pocket pair). For each of those, there are six possible combinations of the suits that could constitute a single-rank flop. That's 66. I'm going to neglect the other two ranks, because even if the other three come on the flop, it's not going to be quads on the turn as shown in the example hand in question.

66/19,600 = 0.00337, so about three out of a thousand times you'll see a flop of a single rank.

What about the turn being the fourth one? There are now five cards accounted for, so 47 left in the deck. Only one of them fits the bill, so only one time in 47 that we have flopped trips on the board will we then see quads on the turn. 0.00337/47 = 0.000072. Inverting that, the result is that only one time in about 13,958 hands will the cards turn out to have four of a kind on the board at the turn, as shown above.

That's rare enough to be worth gawking at for a moment or two, I think.