Sunday, February 22, 2009

Flopping the nuts with poker's best hand

This evening I saw yet another flop of A-3-5. Too bad neither I nor anybody else was holding the holy Deuce-Four at the time. It seems that I've been seeing that flop a lot lately. It could be an illusion of selective attention/memory, because I'm so aware of the 2-4. It made me wonder how often one could expect to see that flop when holding the magic hand. (The only other way that 2-4 can be the nuts on the flop is if it comes 2-2-2 or 4-4-4. Those are rare enough that I'll disregard them.)

Once we have removed two cards from the deck, there are 50 left from which a flop can be formed. There are 19,600 ways to pick three cards from these 50.

There are 4 x 4 x 4 = 64 different ways of making A-3-5. Of those, four have all three cards from the same suit, in which case 2-4 offsuit (which is all I'm considering for now) for the straight is not the nuts.

That leaves 60 out of 19,600 times that 2-4o will flop the nuts, or about one time in 327. It's the same frequency if your own two cards are NOT the 2-4, as long as you don't hold an ace, trey, or five.

I should, on average, see that flop less than once each session. Sure seems like more than that lately, though. Maybe I'm hallucinating. (Or maybe I have, once again, screwed up the math in some embarrassing way.)

3 comments:

Willrr said...

You make so few of these mistakes that I want to point this one out, who knows when I'll get another chance? 2-4 can also flop the nuts on a 4-4-2 flop.

Rakewell said...

Right you are. Thanks for the catch.

Cardgrrl said...

True confession time! I finally won a hand with the mighty Deuce-Four.

Herewith the syllogism:

A) I limped into a cheap pot from the button.

B) I flopped bottom pair.

C) Checked around to me. I bet out 2/3 the pot and everyone folded.

QED: The Deuce-Four is unbeatable.