Combos 101

A deck of cards consists of 52 cards. When dealing a hand in Texas Hold 'em every player gets two cards, and there are precisely 1326 combinations - "combos" - of starting hands.

This is because the first card is any one of the 52 possible cards, and the second one is any one of the 51 remaining. We multiply 52*51 and get 2652 combinations, but in practise there's no difference between and , so we count them as the same hand, halving it from 2652 to 1326.

Knowing that there are 1326 combos of starting hands is not particularly interesting. What's of interest is the fact that understanding combos means we can infer some juicy stuff about what our opponents are likely to be doing. By developing reads and/or using analytical programs such as Hold'em Manager and PokerTracker, we can have some idea of how our opponent plays certain hands. By looking at the board, we can then make some intelligent guesses as to how likely he is to beat us.

To give a practical example, I want to start by showing what happens when we don't use combos:

$100NL, 6max hold 'em. We open preflop to $6 with on the button, and a tight opponent makes it $13 in the small blind. The big blind folds.

We know, from past play with this opponent, that he only 3-bets his premium hands. This means QQ, KK, AA and AK. Because we know his habits so well, because we're in position and because we are getting a decent price on trying to flop a set and stack him, we call.

The flop is

Our opponent shoves his entire stack in. We know exactly what this means. Somehow, we know that means that he has either QQ, KK or AA, but not AK, because he would make a smaller bet with AK. Should we call?

On the surface, it may seem like there are more hands that beat us than that we can beat. He shoves AA, KK and QQ, and of those three, we can only beat AA. But this is misleading, and this is why combos are important:

Before the flop, the likelyhood of him having KK instead of AA was the same. But after the flop, this is no longer the case, because there's a K on the board. That's one less king that he can have in his hand. Looking only at the pocket kings, before the flop he could have any of these hands:

K♥K♦
K♥K♣
K♥K♠
K♦K♣
K♦K♠
K♠K♣

... but now that the is right there, on the table, he can no longer have any of the combos that include that. Removing those, that leaves us with:

K♥K♦
K♥K♠
K♦K♠

... only three.

We can then use the exact same argument with the pocket queens. Originally, there were six pocket queens that he could have, but now that a queen fell on the flop, there are only three left. What about aces? Well, there's no ace on the board, so all six combos of aces are still "available" to him, and we can therefore conclude this:

He is just as likely to have a hand that we beat as a hand that we don't beat (6 combos of each). Because of the dead money already in the pot, we should therefore call.

Another, slightly more practical example, is when your opponent goes all-in on this board on the river in a big pot:



... and you have

Your opponent could be bluffing. He would also shove with a set.

The only hand that beats you is KQ. And you know (or I do, and am about to tell you) that there are 12 combos of KQ that he can have. Originally, there are 16 combos of KQ (king of hearts which each of the queens, king of spades which each of the queens, etc.), but you have one of the queens in your hand, so he can only have three queens for each of the four kings = 12.

12 combos beat you. How many combos do you beat? Well, the three sets that he could have flopped are "3 per set," i.e. three combos of jacks, three combos of tens and three combos of nines, for a total of 9 combos of sets. Furthermore, he could have the lower end of the straight - 87 - which for the same reason as with KQ there are 12 of.

He can have 12 combos that beat you, and you beat 21 (9+12) combos. And that's not even accounting for the fact that he could be bluffing. Hence, you have an easy call.

So when doing hand analysis, we don't just have to look at "what hands" he can have, and how likely he is to play a certain hand, but also how likely that hand is combinatorically. If he plays AA and KK the same way, and there's a king on the flop, he's twice as likely to have AA than KK - which could be very useful for you to know if you have a lower set.

Short summary of various types of hands (not accounting for cards that you know he can't have):

Preflop:
Pocket pair: 6 combos.
Suited connector (e.g. 87s): 4 combos
Offsuit connector (e.g. 87o): 12 combos
Any hand that doesn't have to be suited and isn't paired (e.g. "KQ"): 16 combos

Postflop:
This is tricker because it assumes stuff about our range. But let's make it easy for me and say "only pocket pairs or broadway cards"
Example flop 1 (dry flop): :Qd:
Sets: 9 (three QQ + three 77 + three 22)
Two pair: None (since Q7 isn't in his range to begin with, neither is 72 or Q2)
Overpairs: 12 (6 KK + 6 AA)
Top pair: 48 (AQ, KQ, QJ and QT, 12 each)
Pairs between 77 and QQ: 24 (eights, nines, tens and jacks, each six combos)

Example flop 2 (wet flop):
Sets: 9
Two pair: 9 (QJ; three queens can mix with three jacks = 3*3 = 9)
Top pair: 36 (AK, KQ, QT, 12 each)
Open-ended straight draws: 16 (KT)
Gutshot straight draws: 16 (AK)
Flushdraws: 3 (AhKh, AhTh,KhTh)

Note: Flushdraws seem rare in this scenario. That's because they are in general. Lesson? Not everyone has a flushdraw every time the flop is twotone.

Discussion and questions are of course welcome.

Edit note: These examples are a bit cumbersome to put in - let me know if you want others added.

Pokerstove > Fancy FP maths

:P

Nice post tho.

5 pokerstars (I think we should start rating threads btw).

This is the theory behind the pokerazor analyses I do. And the pokerstove calcs c9 mentioned.

Could you add a postflop hand combos summary? (sets, one pair, two pair etc)

Edited OP to added two postflop examples. Let me know if you want more.

Hall of fame post, imo. Thanks, Fredrik.

Also, c9, fancy maths > pokerstove because it helps you understand the essence of the problem better which is what you need to develop the correct intuition to take good decisions at the table.

Yes, but PS is just so much easier.

And I'm not sure I find the concept all that useful in practice. Like, everyone knows that sets are kinda unlikely, top pair is more likely, ect. And PS uses these ideas to calculate equities & such. But when I'm sitting at a poker table, looking at a flop, I'm not thinking about what kind of hands he can have combinatorically. I'm just narrowing down his range based on his actions.

This post gets to the heart of poker which is "How much equity do I have in this pot & how do I stack up against his range" but for me a lot of this stuff is just intuitive now. Didn't wanna come off mean, or say that this post isn't good. Everyone should read it & take note of the ways hands are distributed. But you certainly shouldn't take this and go "oh, sets aren't likely, and neither are flush draws, so I can get all in here".

No limit poker is 100% about making hands that are statistically unlikely & making your opponents pay to see them.

Yes, and it's become intuitive because you understand the mechanics behind pokerstove's numbers.

Great post, read it once going to read it again. Combos are something I'm not mastering at this point and want to off the top of my head.

Awesome, FP. You and Chuck should be signed up by CC to make one of these posts once a week imo!

Not sure CC can afford to match Chuck's $$/hour at the tables...

I'm sure Chuck would give CC a dizzy!

Any instructional post by Fredrik = must read.


Your Qd in the postflop example is missing the graphic for the d.

Now this is new and great stuff....

Could take some time to analyze how many combos we are likely to beat or be beaten with, if you have the software could become handy.

You made some good examples when you had the nuts or really good hands, but what about when you have absolutely nothing say....

your hand: flop:


how many combos we beat?, how many combos we are beat??
what is he likely to have???

Interesting reading but honestly is this fact/theory really useful?

I know that this is just an example but I mean that if you´re going to deep-analyze a hand and stretch as far as counting combos the probability of your opponent holding AA should be just as big as if he was holding AK since both hands are in your opponents "3-bet-category" and if your tight opponent pushes AA on this flop he will push AK as well since any hand that can beat the AK at this stage also beats AA...therefor you might wanna consider the possibility of your opponent holding a set a little bit higher as you said that he would probably not push AK and IMO not AA either.

To make thoughts a bit clearer let me put it this way:

In your example you eliminated the possibility of your opponent pushing AK, I think that if you eliminate that hand you should also eliminate AA and where does that leave us?

6 combos that we can not beat and 0 hands we can beat!?

This is just a thought and I don´t mean to be critical I´m only trying to clear my head :-) as this way of looking at a hand is new to me

The example was obviously manufactured to demonstrate a point. I could have used a more realistic hand, but that would have come at the cost of having the ranged debatable - and a debate on how an opponent plays certain hands is outside of the scope of this thread. I want you to learn the idea of combinations.

But if you want a more realistic example, try this:

$100NL, full stacks. You have AK. Your opponent raises preflop to $4, you raise to $15, he re-raises all-in. We expect this to be QQ, KK, AA or AK, and neither of them more likely than the other (please don't debate this point, because it isn't important for this thread). Should you call?

You can chuck it into PokerStove and find out, but you can also think about card removal effects. Yes, AA and KK and QQ are all as "likely" in the sense that he would play them the same way, but because you have an ace and a king in your hand, that makes him only half as likely to have AA as QQ.

Or you have AJ on a J-6-6 flop, after raising in middle position and being called by the tight player on the button. You bet the flop and your opponent raises. Are you beat? How many hands beat you?

I think you have to reverse the order of your questions first. Use the preflop actions and the level that you're playing at to determine the preflop range before you do anything else.