Practical use of Combinatorics

For anyone not familiar with this, combinatorics is a big word for a very simple concept in poker. If we can put someone on a range of say AA, KK, QQ, AK then it looks like 75% of the time they’re going to have a made hand where in actual fact only slightly more than half of the time are they going to have better than A high if no A or K hits on the flop.
We work this out by realising that there are 16 ways we can make up AK: AcKc, AcKd, AcKh, AcKs, AdKc....etc
AA has 6 combinations: AcAd, AcAh, AcAs, AdAh, AdAs, AhAs
KK and QQ also have 6 combinations.
So instead of 3 out of 4 hands being big pocket pairs (75%), it’s actually 18 out of 34 which means they’re about 53% to have a pocket pair.
We can do the same thing with the benefit of card removal post-flop, let’s say a villain raised pre-flop and the flop hit ATT, especially the ace hits the pre-flop raiser’s range really hard. Well, how hard does it really hit?
If we assume that from whatever position it was he raised in whatever table dynamics, that his range is something like AT+, KJ+, 22+, 56s-JTs. The A’s in his range are greatly reduced by the board, AK-AJ only have 12 combinations each and AT only has 8 due to card removal (the quick calculation for this is say for AK, you multiply the number of unseen A’s by the number of unseen K’s, 3x4, even I can do those heavy calculations pretty quick).

AT 8
AJ 12
AQ 12
AK 12

KJ 16
KQ 16

22 6
33 6
44 6
55 6
66 6
77 6
88 6
99 6
TT 1
JJ 6
QQ 6
KK 6
AA 3

56s 4
67s 4
78s 4
89s 4
9Ts 2
JTs 2

Pair of aces or better: 52 combinations
KK or less: 110 combinations


What does this mean?
It means that if he cbets 100% of the time that if you’re not floating 100% of the time, you must not like money very much.
If he cbets 50% of the time... well, I have no idea how you should proceed based on combinatorics, that’s kind of the reason for this thread to hopefully create some discussion about this stuff to hopefully make this more clear.
Keep in mind that as soon as he takes an action that he wouldn’t make with 100% of his range his range has to change after that action. The actual hands in his range don’t change, just the frequency that they will show up. For example the tens in his range include only 13 combinations out of 162 but if he bets into us all the way to the river and shoves those 13 combinations become a very significant portion of his range. However we never completely lose even 56s from his range but its value reduces to almost nothing, it will show up at showdown occasionally but the frequency that he decides to bluff three streets with it is presumably low.

The nearest I can come to a conclusion about combinatorics is that it’s incredibly useful but the more actions that take place, the lower its accuracy becomes and the harder it becomes to calculate in real time. The best method seems to be estimation, in the above ATT board example, assume TT not to exist until given significant evidence that he has it, the same to a lesser degree with various other Ts. By the time you get to the river, well, I’ve made enough dumb decisions based on this stuff that to be honest I think the most you should usually apply it is to consider that unpaired pocket cards are much more likely than paired and otherwise ignore combinatorics and focus instead on the rest of your poker decision making considerations like board texture and player tendencies.

Any thoughts?

Just to add to the above because obviously things are easier to figure out AFTER you've made all of your statements/questions/whatever.

In the 50% cbet example, he's cbetting half of the time and has a real hand roughly 33% of the time so yeah floating with any two cards is fine but you have to watch out for the top of the air portion of his range having decent equity.

That's what I meant.

vn post!

I only use combinatorics and card removal when faced w a close decision and Villain is repping a narrow range - I'm sure there are more uses for it than that. Applying it to cbetting ranges isn't something I'd considered - I pretty much look at cbet stats and board texture to determine whether I want to continue - looking forward to the discussion.

It works well even for decisions on later streets as long as their range will play the same way, as soon as there's any reasonable frequency that they play part of it differently combinatorics effectiveness starts to break down, imo.

With that being said, I only very recently started relying on it more on early streets and less on later streets so could really easily be wrong.


It's a widely used concept for a lot of situations. It's purpose is to give you a better picture of what you're up against rather than just a range of hands you get a better estimate of the frequency of those hands within a range. For example AK being nearly 3 times more likely than AA is a better estimate than considering them both to be just one hand in a range, but AA is also never getting folded pre-flop where AK will occasionally hit the muck so we're still dealing with estimates.

I believe you misread.
It's 65 quality posts.
Good luck with the freerolls though.

Hmm, hand from a session yesterday - kinda thinking through practical applications, does it make sense?

Poker Stars $0.50/$1 No Limit Hold'em - 6 players
The Official 2+2 Hand Converter Powered By DeucesCracked.com

BB: $111.75
Hero (UTG): $129.95
MP: $103.00
CO: $135.30
BTN: $116.75
SB: $20.50

Pre Flop: ($1.50) Hero is UTG with Q♥ K♥
Hero raises to $3.50, 4 folds, BB calls $2.50

Flop: ($7.50) 6♣ K♦ Q♠ (2 players)

BB checks, Hero bets $5, BB raises to $10, Hero raises to $28.25, BB raises to $46.50

Villain is 26/18/2, 4.2% 3bet, over 150 hands, and I've suspected him of playing back at me light (not on this table, on other tables).

OK, so Villain's repping a strong hand obv. Card removal and pf suggest he should never have KK or QQ here. Very few combinations of 66's. No flush draw, so the only other holdings would be straight draws, Kx hands, and air.

While I suspect him of playing back at me light, this isn't a spot where that makes a ton of sense (oop, I'm utg opener, KQ hits my range very hard - I don't think he's an idiot).

What do you think?

66/JT (3 compared to 16 combinations) and expect to see KQ more often than either KK or QQ (KQ is more likely to just call pre-flop than KK/QQ).

Except if he's semi-competent then he'd know his raise sizing is terrible for JT and only make sense for 66/KQ/KK/QQ or something like AJ that he plans to dump if raised. So I fall back on how good I think he is, if he knows what he's doing I fold and if he doesn't I shove. I shove almost always.

Only 2 K's and 2 Q's left...

I actually saw this response the other day and couldn't figure out how to respond to it without stating the obvious.

There's 2 K's and Q's.

His range pre-flop against an UTG raiser would usually be around AK/AQ, KQ, 22+ and the occasional SCer (since he's playing back at you a bit) isn't a terribly inaccurate range, and for the purposes of an example, it will do.

Before any action on the flop, that's 64 combos that you beat and 5 that beat you and 4 that you tie with.

By the time he 4bets the flop, AK/AQ will show up occasionally as bluffs, 22-JJ will individually show up almost never but combined I guess he'll bluff a little with them. The hands that you beat now make up a smaller part of his range. Like I said in the original post, I don't think hand ranges ever change it's just the frequency that hands will show up that changes so his range is still 64 hands that you beat and 5 that you don't, but the 64 is worth a lot less now.
They just wont show up as often so treating every unit of 64 the same as you would treat each unit of the 5 (which always take this line) doesn't make sense.

This is why I don't like relying on combinatorics after many actions post-flop.

More actions = more opportunities for villain to play hands different ways.
Which makes any statistical analysis way too complicated to do in real time.



I think I may be butchering the subject of "Practical use of combinatorics", but whatever.
Before any action is taken every hand has an equal chance of showing up, the more action that gets taken the further each combination is from it's initial value.

If anyone wants to add to this thread (or just explain how I've got it all wrong, if you'd like) before I over-complicate combinatorics further, please do.

I'm a little late to this thread, and I actually don't have time to post too much right now, but a few things that I think everyone should be able to answer immediately upon being woken up in the night are things like these:

1. How many overpair combos on a 9-high board?
2. How many combos of sets on a X-Y-Z board?
3. How many combos of sets on said board if I have two pair?
4. How many percent of someone's opening range consists of pocket pairs (assuming they open all pocket pairs in this position)?

6tabling lurkscribe

Hello everyone! Very nicely explained! There seems to be one mistake though. AT is only 6 hands on ATT board (2 unseen T's times 3 unseen A's).

I'm game

There are actually 5 combos of sets if you have two pair.

How can you actually answer the 4th question? To me there is not enough information...

Sorry if Im wrong since Im still trying to wrap my head around this but should there be 9 possible sets for question 2 (3 different pocket pair combinations of X, Y, and Z = 3x3 = 9?)

I think for question 3, there should only be 5, (3 for the one with no two pair and 1 for the two pair. If I have XsYc and a XhYhZh board, the combinations of sets my opponent could have must be XcXd, YsYd, ZcZd, ZcZs, ZsZd). Im not sure how we get answer four.

Im still not understanding the practicality of this since it seems to work only if we are able to put villains on accurate ranges. I think this can help us predict whether we a really beat or not, that given our hand and the range of our villain, of the hands villain would probably limp in vs raise in this situation, we can assume villain has one of x hands and in that range, villain could have these hands which beat us while he could have these hands which dont. Although this is helpful to know since I currently use worst case scenario probably give too much credit to my villains, I feel that this is only useful once we set an accurate range on our villain which I find nearly impossible to do in a tournament due to the changing blind structure forcing people to open.