Combinatorics (very very long)

[derosnec]

Chomp asked me to post about counting combos, so I have some time now.

For some (or most, I don’t know) this post might be obvious and second nature. But maybe for a few it will be helpful. While reading this, keep in mind that I have an IQ of about 20 or 30 in the mornings and that I'm not a coach or anything, so if there are errors, please tear into me all you want. I like to learn.

Combinations 101

It’s easy to do. Take the remaining cards and multiply them by each other. Check Poker Stove to verify.

Example: Flop is A67. You have QQ.

How many ways can the opponent have AK? 3 aces remaining multiplied by 4 kings remaining = 12 ways he can have AK. So 48 ways for AT+.

How about 67? 3 * 3 = 9 ways.

What you will notice after doing this after while is you’ll start to gain some insights. For example, there are a lot of straight draws on many flops. On A67, there are 48 ways he can have an OESD (16 ways of 98, 16 ways of 54, 16 ways of 85) 64 ways he can have a gutshot (16 ways of T9, 16 ways of 43, 16 ways of 84, and 16 ways of 95). That’s 112 ways he can have a straight draw (ok, 95, 84, and 85 aren’t very realistic, but you get the point).

Let’s try AA on a QJJr flop. You are the pfr and bet. He calls.

Are you generally ahead here?

Assume we put his pf range at 22+, A8+, K9+, Q9+, J9+, T8+, 98, 76.

The first thing you’ll notice is the ways he can have top pair. AQ = 6 ways (because 2 aces left and 3 queens left); KQ = 12 ways; QT = 12 ways; Q9 = 12 ways. So that’s 42 ways he can have TP.

What about straight draws? 32 ways he can have an OESD; 8 ways he can have AT; 8 ways for AK; 16 ways he have T8. And 16 ways he can have K9. 16 ways for 98. That’s 96 ways he can have a straight draw. Impressive, huh?

And trips or better? 4 ways he can have AJ; 8 ways he can have KJ; 3 ways he can have QQ; 6 ways for QJ; 1 way JJ; 8 ways for JT; 8 ways for J9. 38 ways he can have trips or better. That’s almost as many ways as he can have top pair.

The other hands: we’ll assume he folds 22-88/A8/A9 to the flop bet. So, that leaves 6 ways for KK, 6 ways for TT, and 6 ways for 99. 18 ways total.

Add it all up and we get:

We’re ahead of 156 combos.

We’re behind 38 combos.

Range versus Combinations

The example above is a bit more revealing (to me at least) of the reality of opponent’s distribution of hands than merely just saying he has X range.

If you are only using ranges, you are – to use an analogy - looking at a piece of bread, while those who use combinations are looking at that bread with a microscope and seeing all that constitutes it (maybe that’s a dumb analogy).

Reads versus Combinations

You might argue, “But I have my reads, man. I don’t need to count no stinking combinations.”

Reads, in my opinion, have severe limitations and are not as good guideposts in assessing the strength of our opponent’s hand postflop as we think they are.

The reasons why might be obvious but I’ll discuss them anyway.

1)Sample size: Say you have 300 hands on an opponent (which is a lot of hands against a micro opponent since we are not playing the same people often, as opposed to MSNL/HSNL). Assume that opponent has a VPIP of 20 and WTSD of 25. That means – at best – you saw 15 showdowns where he put money in preflop (not checked his BB). So you have direct hand evidence of 15 hands. What do 15 hands tell you? Very little. See the next sections for why.

2)Preflop dynamics. Sometimes an opponent is the pf raiser, sometimes he is the caller. This affects how he plays his hands postflop. In other words, he might play TT different as the pf caller on a Qh6s7s flop than he would as the pf raiser.

3)Number of opponents posftlop: How an opponent plays top pair, a draw, or a monster multiway might differ how he plays it heads up.

4)Flop texture: There are roughly 17,000 flop combinations heads up (I think). Our reads are based on a tiny % of those flops that opponent has seen. Take opponent having a set for example. Will he play it the same on all flop combinations? Our read on how an opponent plays a set will probably come from only one or two flop combinations observed.

5)Position: How an opponent plays a hand out of position, on the button, or in good/bad relative position will likely affect how he plays his hands postflop. So, he might play a set different OOP than he would on the button or in good versus bad relative position.

6)Steal situations: An opponent might call more/be more aggressive in these situations than in non-steal hands.

7)Your image: This is very important from my experience. When I tightened up over my last 100k hands (my vpip dropped to about 14 or 15), I felt like I was getting coolered a lot. Maybe I was. But what was probably happening is opponents were playing their hands differently against me than they were against other opponents. So, while an opponent would play TPGK fast against another player, he was playing it slow and cautiously against me, but playing his sets fast against me.

General Micro Plays/Theorems versus Combinations

A lot of players at the micros make the same plays. They minraise with sets, slowplay the flopped nut flush, don’t c/r bluff the river, put you on AK, don’t fold full houses (Zeebo Theorem), raise turn with better than top pair (Baluga Theorem), etc.

So, for these situations, combo counting is of less value.

The Not-So-Obvious Value of Combinatorics

So, assuming you’ve made it this far in the thread, you might be thinking, “No way in hell am I doing all this counting at the table. I don’t have a time bank of 2 hours per hand.”

You want to review your hands in PT and apply combo counting to see if you made good decisions. After you do that for a while, counting will become much quicker and intuitive because you will have seen so many similar scenarios. So, counting top pair combos on the flop, for example, will require little thought.

Now, given this practice, you will start to gain quite a few insights. When you begin to apply these insights, you will become a much tougher player. What will begin to happen is that your game will start to lean towards a Game Theory-ish Optimal strategy.

True, you are exploiting your opponents based on the ways they can have a hand, but by doing that you start to make your plays less based on your own hand and more against the distribution of your opponents’ hands, which makes your opponent's task of exploiting you very very difficult.

An example:

So, after spending way too much time analyzing PT hands, I noticed that I was doing a lot of this:

tag UTG/MP raises pf. I call with a small pp for set value.

Heads up, the flop comes A62r. I check, he c-bets. I fold having not made my set.

But after playing around with combo stuff, I determined that they do not have a pair of aces or better on that flop the majority of the time. So I started to check-raise here or raise from LP (doing it in a way so that it is profitable based on pot size).

Now I was no longer playing my hand. Of course, if I do this 5 times in a row against him he will likely adjust, but I can flop sets/two pair/etc too.

And on turns and rivers, your play will naturally become much tougher. If I end up on a river, and given opponent’s play so far I assess that there are 70 ways he can have a better hand and 40 ways he can have a worse hand, but only 24 ways he can have a monster, I can play aggressive enough to cause him to fold more often than he should.

Also, by playing with Poker Stove, another insight you will likely gain is how few legitimate flush draws an opponent can have compared to other hands. For some reason our brain spots flush draws right away, despite the reality that they are often a tiny part of the ways he can have a hand.

Weighting

Just because someone has a range of XYZ, doesn’t mean X is as likely as Z. So, in Poker Stove, you go through a hand and reduce the ways he can have a hand so you can weight them appropriately. So, if KTo is in his range, but rarely so given pf and flop action, I’ll reduce it from say 12 combos to 2 combos by only selecting KcTd and KhTc, for example.

Warning

At 10nl or lower, where your opponents are rarely engaged in the process known as "thought", be very careful about ever bluffing. At 25nl, there are a few tags, and at 50nl, there are alot of tags who do think. So, keep that in mind.

To see combo counting in action by one of th ebest, look up Bobbo Fittos' posts. This a good one (his analysis is towards the end of the thread): Link

[bozzer]

w00t!

[matrix]

[img]/images/graemlins/heart.gif[/img] derosnec - good stuff.

[tms]

wow awesome stuff, going to study/work on it right now

[Bonesy]

Finally someone explained the combo platter. Thanks a lot!

[poisonxfree]

Awesome. Bookmarking this for next week when I can actually start playing again.

[Waingro]

Lol I feel so retarded after reading posts like this. Clear, easy to read and very helpful. I am more in the "LOLZ I have aces I push" camp, but I will try to use this approach sometimes as well. Especially when reviewing a hand in pt.

Sticky plz obv.

[Fiksdal]

tldr

I certainly will though, during one of the boring classes this week. looking forward to it

[Chomp]

Jesus, about time.

[Bowlboy]

Nice post. I gave this stuff some good thought awhile ago by pulling out a deck and figuring out the number of combinations for non-paired hands though I have to admit I have'nt practiced counting them after sessions as you suggested and that is a great idea.

I just wanted to add for beginner combo counters that for pocket pairs the total possible combinations is 6 ways. I dont know what the calculation is, I believe it's a permutation though not sure, I'm useless at math. So if you want to know how many combinations are possible for AA it's 6. If there is an Ace on the flop then there are 3 aces left and that also means there are 3 combos. With 2aces out there is only obviously 1 combo. So far pocket pairs just remember 6,3,1.