Wednesday, January 26, 2011

Sometimes you feel like the nuts, sometimes you don't

It happened twice over the weekend, exactly the same way both times. There was a showdown, one person had suited A-x matching the suit of three of the board cards and winning the pot, followed by an uninvolved player commenting, "Just the nuts."

Problem is, it wasn't the nuts either time. Not even close. In both cases the board was paired.

I remember the first time I ever played poker in a casino. It was at the Luxor, in about 2003, maybe 2004. It was just a stupid daily afternoon tournament, buy-in of about $40. I lasted all of 20 minutes. I went out when I had a flush. I was surprised when I didn't win the hand. The other guy had a full house.

"Well, how was I to know that he could have been that strong there?" I thought. At the time, my knowledge of the game was so crude that I did not understand that a full house was a possibility anytime there was a paired board. (I use this, my own case, as a prime example every time somebody says something about the hand in progress that they think is so obvious that everyone already knows it and so it could not possibly affect the action. You just never know how rudimentary some other player's understanding of the game is, and, therefore, what may or may not help him or confuse him.)

I later worked out on my own that a paired board meant that exactly six different full houses were possible. I drilled myself over and over again until I could spit out, rapid-fire, what they were, in descending order of strength.

When the Hard Rock briefly experimented with the game variant called Royal Hold'em, they didn't have a poker room, and thus had to quickly retrain some blackjack dealers, and they didn't know poker well at all. One time I was sitting there by myself hoping that some fish would come to play, got chatting with the very attractive young woman dealing, found out how shaky her poker knowledge was, and spent the time improving her showdown hand-reading facility. She was deeply impressed with my being able to rattle off all the best possible hands in order. Ya just never know what it will take to pick up women!

Hmm. I seem to have wandered off-topic here.

Anyway, the repeated non-nuts "nuts" this weekend got me to wondering how many possible hands there could be ahead of the nut flush with different types of boards in hold'em. I'm going to take as my slightly silly hypothetical that we're playing at a 22-handed table (thus using the entire deck, after accounting for five board cards and three burn cards).

The easiest scenario is when there are quads on the board. That's easiest because it's not possible for anybody to have a flush, so the question is moot. The same is true if there is a full house on the board.

If there are trips on the board, then obviously one person can have quads, and anybody holding a pocket pair has a flush-beating full house. Suppose the board is Ac-As-Ah-Kh-7h. The nut flush is Qh-Xh. But that poor sap is being beaten not only by somebody holding the last ace, but by anybody holding any pocket pair, and even by anybody holding any K-X hand or any 7-X hand.

In theory, the nut flush could actually be the worst hand among the 22 players at our very large table, given the board listed in the previous paragraph. As just one way of dividing up the deck, let's suppose the three burn cards are all deuces, as is the kicker for the nut flush. We could have three players with K-3, three with Q-3, two with J-J, two with T-T, two with 9-9, two with 8-8, three with 7-4, two with 6-6, and two with 5-5, giving all 21 opponents a hand that beats the nut flush. [Edit: As pointed out in the comments, I made a mistake here. Here's my second attempt: one with A-3 (for quads), three with K-3, one with Q-Q, one with Q-7, two with J-J, two with T-T, two with 9-9, two with 8-8, two with 7-4, two with 6-6, two with 5-5, and one with 4-4, and we've got our 21 hands that beat the nut flush.] If we ignore which cards can be held simultaneously and just rank-order the possible hands, the nut flush is actually the 24th nuts, falling behind 12 different quads hands (A-K down through A-2), and 11 different kinds of full houses (aces full of kings down through aces full of deuces).

But wait, there's more! If I optimize my board to, say, 9h-8h-7h-7c-7s, then the nut flush drops to the theoretical 30th nuts--behind three different ways to make a straight flush, 12 different ways to make quads, and 14 different ways to make a full house (9s full one way, 8s full one way, and 7s full 12 ways).

Suppose the board is double-paired, e.g., Ah-Ad-Kh-Kd-2h. Then there will always be a theoretical minimum of two players who can beat the nut flush--those holding quads. If nobody has quads, then there can be a maximum of five full houses lurking among the opponents (e.g., A-7 times two, K-7 times two, and 2-2).

But we can do better than that. Let's instead made the board 9h-9d-8h-8d-7h. Now the nut flush is Ah-Xh, but it can be beaten not only by up to five boats, but also by any of three different straight flushes, and two of those can exist simultaneously, for a maximum of seven players at once crushing the nut flush (e.g., 9-2, 9-2, 8-2, 8-2, 7-7, Jh-Th, 5h-6h).

Now let's look at the kind of situation that pertained in both of the examples I started this post with. The most ordinary kind of paired board will have one set of quads and six different kinds of full houses available, making the nut flush only the 7th nuts. How many players might simultaneously be sitting on hands better than the nut flush? Five. Suppose the board is Ah-Ad-9h-5h-2c. We can have, e.g., A-9, A-5, 9-9, 5-5, and 2-2 out there. So in our 22-handed game, Mr. Nut Flush can at least have the small comfort of knowing that he will be in no worse than sixth place at showdown.

But again, if we tweak the board to full ridiculousness, such as 9h-8h-7h-6h-6c, the nut flush is the 10th nuts, standing in line behind three different straight flushes (jack-high, 10-high, and 9-high), one set of quads, and six different full houses. How many of these could exist at once? Seven, maximally: two straight flushes, and five full houses. Which means that in this scenario, Mr. Nut Flush might be only eighth-best at the showdown.

Even on an unpaired board, with no quads or boats on the horizon, the nut flush might not be the nuts when a straight flush is possible. Who here has lost a stack to a straight flush while holding the nut flush? (Grump raises his hand.) No more than two players can have a straight flush at once, so at least when the board is unpaired, holding the nut flush means you have at least 19 of your 21 opponents beat, and that's not bad!

All of which is an exceedingly long way of saying this: Not only is it an error to equate "nut flush" with "nuts," but sometimes the two things are a very, very long way apart. As shown above, it's possible for the nut flush to actually be only the 30th-best possible hand for some kinds of boards. If you call "nuts" when there are 29 hands that can beat you, you're due for a serious dope slap.

(What are the chances that I've made it through all of that without having made a mistake at some point?)


Justin said...

Unless you usually play with six 3s in your deck... In you first example you list three K-3, and three Q-3 hands. (I'm not even sure why Q-3 is any good there in the first place)
Chances of a mistake are good I guess ;-)

Michael said...

I'm going to come back and read the breakdown again once my eyes uncross, but I wanted to say thanks to you as I was in the same spot as you (at luxor tourney) when I first started playing, in that I just didn't have the handle that a pair = possibility for a full house. It wasn't until I started reading your blog (and subsequently playing more) that I put it together.

Chris said...

One possible error: How does holding Q-3 on a board of A-A-A-K-7 beat the nut flush?

John said...

You have too many threes in your deck for the first example and although it's early here for me I don't see how Q3 beats the nut flush there. It doesn't change that you can create 22 hands that beat the nut flush, but I don't want to be playing with a deck with extra threes. I get those too often already.

Bouza In Britian said...

Also again, I'll start off with a thanks for the post. Even now I find myself missing the chance of a boat when I get too distracted by my own hand.

And just trying to help for clarification sake, your example of A-A-9-5-2 contradicts itself. First you say nut flush would be 7th, but then later miss the A-2 and state the nut flush would be trailing 5 hands.

Love the blog and also being able to introduce myself quickly as a reader one September Sunday at Mandalay. Didn't want to bother you further, so I went on my way.

Rakewell said...

BiB: It's not a contradiction. The nut flush is the 7th nuts if all you do is look at what hands might theoretically beat it. But not all of those hands can exist simultaneously, because they use each other's cards. A maximum of 5 of them can actually be in opponents' hands at the same time.

Bouza In Britian said...

I see what you are saying. Thanks for setting me straight.