Combinatorics in Texas Hold’em Limit Poker

Combinatorics is the mathematical technique of enumerating and analyzing combinations of objects.  If you've ever had a pair of pocket Aces preflop and had an opponent bet his whole stack against you, you did a form of this analysis in your head.  Presumably your reasoning went something like this:
"I have the best hand.  Any two unpaired cards or any other pair is an underdog to me.  On the rare chance that my opponent has Aces, we're at best dead even in probability to win this hand.  Over all those hands, I'm an overall favorite, I shove."

Combinatorics are not necessary in that situation since the math is obvious.  But what about later in the hand, when you've got something less than the nuts and you've got to examine the range of hands that intersect with the common cards on the board?

When you don't have the nuts, knowing where you stand is important.  When you're a solid favorite, without the nuts, you should still raise when you're a favorite versus the likely range of your opponents hands.  Here's an example from a recent Limit Texas Hold'em $.50/$1 heads up game I played at Pokerstars.   

Ok, onto the hand.  I'm the Hero, my opponent is the Villain, obviously.

Preflop: Hero is Button with
Hero raises, Villain 3-bets, Hero calls

I'm not going to do too much preflop analysis here, but suffice to say that my hand is in the top 15% of all hands, so not only should I raise it, but I can call three bets.  (One might argue that I missed a chance to cap it. I would not disagree)

Let's see how the rest of the hand went down.

Flop: (6 SB)  (2 players)
Villain bets, Hero calls

Turn: (4 BB)  (2 players)
Villain bets, Hero calls

River: (6 BB)  (2 players)
Villain bets, Hero raises, Villain 3-bets, Hero calls

Total pot: $12 (12 BB) | Rake: $0.50

Hero had K, 10 (two pair, Kings and tens).
Villain had K, A (two pair, Kings and eights).
Outcome: Hero won $11.50

Our villain's line is Bet-Bet-Bet/Raise.  (That means "Bet on the flop, Bet on the turn, and bet and re-raise on the river).  What possible hands fit that line and also put in 3 bets preflop?

  • Some combinations of 8x would play that way.  The preflop 3 bet implies A8s, A8o and K8s.  I could also add 98s.
  • A number of pairs would play this way, including 66, 88, TT, JJ, QQ, KK, and AA.
  • Some combinations of Kx would play that way, including AK, KQ, KJ, and KT.  
  • An aggressive player might play 97 this way, betting until they hit their draw.  Definately 97 of diamonds, since it would have an ungodly number of outs with the flush draw.
  • A flush draw being played aggressively this way might bet every street on the come, but wouldn't three bet the river without hitting something, such as an 8, a K, or maybe a T.  I would think we'd have to be looking at a pretty strong draw, such as A8 diamonds, AK diamonds, or maybe AT diamonds.

So that's the range we're up against.  Given those chances, what's my overall chance to win this?  How strong am I?

So check this out, 26 times out of 66 results, I lose.  That's 39% of the time.  36 times out of 66 results, I win, that's 55% of the time.  The board has a straight, three of a kind, and even quads on it, and yet my hand beats most of the hands that would play this same way.  Clearly, I should at least raise and then call 3 bets if it comes back.  Putting in a fourth bet is somewhat player dependent, but I chose not to.

Quite often, you find yourself in this situation and because there are so many hands that beat you, that you check/call or call the river, when in fact you should be raising with a hand that beats most of the field.

Of course, you can also do this the easy way with Pokerstove.  Just load these up options into Pokerstove and run the results:


There are shortcuts you can learn that lets you quickly figure out how many hands are out there, but that's a topic for another post.