Poker Combinatorics: A Simple Approach

It’s a scary word isn’t it? Com-bi-na-tor-ics. So long, and so many syllables. The definition is just as overwhelming, “Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures”. (source Wikipedia)

Relating to Poker, this important concept doesn’t have to be so daunting once you understand the constraints and take the right approach. All poker combinatorics (or combos we’ll call them) really means here is how many times it’s possible to be dealt each type of hand.

Applications of Poker Combinatorics

Any Pocket Pair = 6 combos So there’s 6 POSSIBLE WAYS for any 1 pair to be dealt from the deck.

** Card removal factor: if you assume your opponent may have flopped a set holding 55 on a J75 Flop when he raises you, the number of combos is halved because of the presence of a that 5. There are now only 3 ways he could hold 55.

Any unpaired hand like AJ = 16 combos (4x Aces, 4x Jacks.  4×4 = 16)

** Card removal factor: If we assumed our opponent had A♦J♠ on that same J75 flop, then this also reduces the number of Jacks he can have on a J75 flop. You adjust for this by simply removing a jack from the possible combo count and multiplying again:

4x Aces, 3x Jacks. 4×3 = 12. There are now only 12 combos available for A♦J♠ on a J75 Flop.

Great but what do I do now? You might be asking. Well you do this:

  1. Combos you beat / Total no. of combos (including the ones beating you), gives you a percentage.
  2. Compare this to the price offered by their bet. 1 Needs to be greater than 2 for a +EV call.

Lets do an example using the above board, 100bb stacks NL100. Our opponent is a predictable 40/0 Fish in the Big Blind and only raises with the nuts.

We have AA, we raise to 4x and get called. Pot $8.5bb

The board is J♦7♠5♠.We pot bet a good board for a wide calling range and to our surprise he shoves All-in.

Lets count some combos!

What would a loose-passive fish shove with? Well the clues in the name: loose-passive. He checks and calls his draws, weak made hands and shoves his big hands when they’re vulnerable. Having not worked out the math yet as I write this, AA might be a call here, but let’s see.

Shoves that we beat:

KK, QQ = 12 combos. Remember he’s a 40/0 he certainly has these hands in his range, and he could well play them this way too on a vulnerable board.

Shoves that beat us (20 combos):

Sets = 9 combos JJ, 77 55

Two pairs = 11 combos 75s, 75o, J7s

Combos we beat 12 / total no. of combos 32

12/32 = .37 or 37% we are ahead given our range estimations.

The pot after his shove is $113

Our remaining $87.5/$200.5 (x/x+y) = 43.5% average equity needed for a +EV call. We are only ahead 37% and probably have to fold… Or do we? Comparing how frequently we are ahead with the average equity we need are not the same thing.

In this case when we are ahead we dominate the Villain 91%-9% roughly, and when we are behind they have two pair combos about half the time. In order to actually determine whether this is a call or not we need to sum the equity of the 32 scenarios and divide by 32 to determine our “average equity”.

We know our equity versus KK and QQ is approx 91%. Our equity when we are up against two pair is approx 27% and versus a set is approx 10%. Therefore, to really figure this out we go ((91*12)+(27*11)+9(10))/32=46.37% and that makes this particular scenario a call. While we are only ahead 37% of the time, which at first glance could suggest a fold, once we calculate the average equity we should actually be calling.

How Tow Use Poker Combinatorics

Note: I know you’re thinking “Yeah right, you don’t do poker combinatorics while playing! What’s simple about this!!”. You’re right I don’t, and it’s not simple as such. What IS simple is what’s going to be your approach to combo counting. I’m not telling you to count Combos while you play. I got better at poker combinatorics away from the table, and gradually applied that knowledge at the table. Just like any strategy. That’s how I knew it was a +EV call… I’’ve done the work before. Now you can to.

In general, the wider your opponents range is (i.e lots of pre-flop unpaired hands racking up the combo count) and the more aggressive they are, the better your strong range will do vs theirs. But it takes that work away from the table to build up some intuition for the various situations. Take it slow. Start pre-flop, then a few uncomplicated River calls, and before you know it you’ll have developed that ‘feel’.

Be sure to check out ThePokerCapitalist’s new book Exploitive No Limit Holdem: An Experts Insight on How to Become a Winning Poker Player is now available. Follow @PokerCapitalist on Twitter.

Hope you enjoyed this post on poker combinatorics and I would love to hear your comments!

Cinch

Up next…  A Poker Misconception Quashed

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1 Comment

  1. Thanks \”Cinch\”! Great job on this post

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