Flopping trips and tripping flops

Last night after I left the Orleans I drove to the Riviera and played there for a while. In the first hour, there were three flops that contained all the same rank of card: specifically, 9-9-9, 8-8-8, and 5-5-5. That seemed like a remarkable coincidence to me, and it got me wondering how often this should be expected to occur. Let's work it out.

The first card can obviously be anything. Let's say it's the ace of spades, just for funsies. Now there are two cards left to come for this flop and 51 cards unaccounted for in the deck. There are 1275 combinations of two cards that can be picked from a deck of 51 when we don't care about the order they come in, because C(51,2)=1275.

There are three aces left. There are only three different pairs we could possible select from them: Ad-Ac, Ad-Ah, and Ah-Ac. So only 3 out of the 1275 possible pairs of cards that we might draw to complete the flop will meet our stated criterion of all three flop cards being the same rank.

In other words, this should happen 3/1275 hands, or 1 in 425 hands, or about 0.24% of the time--a pretty rare occurrence, all right.

So I was correct: seeing it three times in an hour was highly unusual, given that a typical hour of live casino poker usually won't produce more than 40 hands or so.

Isn't math fun?!