How-To: Calculating Hand Combinations, About Blockers, etc.

*Last edited by spoonitnow on Wed, 05 Aug 2009, 9:58pm; edited 3 times in total*

List of Topics:

1. Introduction to Hand Combinations
2. How to Calculate Possible Combinations of a Pocket Pair Hand
3. How to Calculate Possible Combinations of a Non-Pocket Pair Hand
4. Brief Discussion of Blockers
5. Effect of Blockers on a Villain's Range with Example Calculations
6. Some Additional Applications of Blockers

1. Introduction to Hand Combinations

There are 1326 possible pocket hands in hold'em if you assume that something like A Spade

A

is the same hand as A Heart

A

, which it is. We get this number by saying there are 52 ways to draw the first card, 51 ways to draw the second card, and divide that product of 52 and 51 by 2 since we're assuming A Spade

A

is the same as A Heart

A

. Statistically, it's more likely for you to be dealt AK than AA, and more likely for you to be dealt AA than AKs, and so quite obviously a study of the combinatorics of starting hands can be useful for developing a more accurate understanding of the development of ranges during a hand.

In this post I'm going to cover what I consider to be the most important topics on the subject. First, I'm going to show you the fastest way to calculate how likely it is for someone to be dealt certain hands. Second, I'm going to show you how to put this into practical use with a discussion of blockers.

I would like to note that this is a lot easier than most people make it, and with just a few minutes of practice, you will be able to tell me in about two seconds that it's equally as likely for someone to be dealt AA or KQ if the flop comes KQQ.

2. How to Calculate Possible Combinations of a Pocket Pair Hand

If you randomly choose two cards from a deck, there are six different ways to make a single pocket pair. If you examine the possible ways you can be dealt 55, for example, you come up with 5 Spade

5

, 5

5

, 5

5

, 5

5

, 5

5

, and 5

5

. For each pocket pair, there are the same combinations of suits.

Intuitively we can figure out that on a flop of AK2, Villain having QQ is more likely than Villain having AA. Similarly, on a flop of 885, Villain having 55 is more likely than Villain having 88. The question that most people can't answer is: how much more likely is one hand than another? To answer this question, I'm going to ask you to remember a number, and it's the only thing I'm going to ask you to memorize in this entire post. The number I need you to remember is: 6310.

Here is how you use this number to determine pocket pair combinations. If we are asking how many combos of AA there are, and we see no Aces in our hand or on the board, then there are 6 possible combinations of AA, just like we stated earlier. However, if there is 1 Ace in our hand or on the board, then there are 3 possible combinations of AA that a Villain can have. If there are 2 Aces in our hand or on the board, then (quite obviously) there is 1 possible combination of AA that a Villain can have. And finally, if there are 3 Aces in our hand or on the board, then (again, quite obviously) there are 0 possible combinations of AA that a Villain can have.

So suppose we hold KT on a board of AAQJ7, and we want to know how many combinations Villain could have of AA, QQ, or 22. We can see two Aces exposed, so there is 1 possible combination of AA that Villain could have. We can see one Queen exposed, so there are 3 possible combinations of QQ that Villain could have. Finally, there are no Twos exposed, so there are 6 possible combinations of 22 that Villain could have.

For another example, suppose we hold 44 on a flop of AJ4. There are 3 ways Villain could have AA, 3 ways Villain could have JJ, 6 ways Villain could have KK, and 0 ways Villain could have 44.

For our final example, suppose we hold AK preflop. There are 3 ways Villain could have AA, 3 ways Villain could have KK, and 6 ways Villain could have QQ. (This ties in with our discussion of blockers later on, but note that it's because of this that AK is only a 40/60 dog against QQ+, AK).

3. How to Calculate Possible Combinations of a Non-Pocket Pair Hand

There are sixteen different ways to be dealt a non-pocket pair starting hand in hold'em because there are four possible suits for the first card and four possible suits for the second card, and we all know that four times four is sixteen. In light of this, it's not complicated to figure out how many possible combinations of a non-pocket pair starting hand there are.

The first step is to figure out how many of each card is left in the deck. For example, if we want to know how many combinations of AK our opponent could have, then we want to know how many Aces and Kings are left in the deck. The second step is to simply multiply these two numbers together.

For example, if I want to know how many combinations of QJ an opponent could have when we hold AJ on a flop of JT4, then I note that there are 4 Queens left in the deck and 2 Jacks left in the deck, so there are 8 possible combinations of left QJ that a Villain could have.

For another example, if I want to know how many combinations of AK an opponent could have when we hold AQ on a board of AKQ53, then first I see that there are 2 Aces and 3 Kings left in the deck. Therefore, there are 2x3 = 6 possible combinations of AK that a Villain could have.

4. Brief Discussion of Blockers

When you hold a blocker, you hold a card that makes it less likely for an opponent to have a specific starting hand. For example, if in hold'em you hold A5s before the flop, it's half as likely that a specific opponent is dealt AA than if you held KQ or KK or 87s. There are other examples as well, such in 2-7 single draw if you are dealt 22277 then it's much less likely that an opponent was dealt a pat 7 (75432, 76432, 76532, or 76542).

This is very intuitive, but the effects of blockers are often misrepresented and misunderstood, which is why I've added the following onto this post about starting hand combinations.

5. Effect of Blockers on a Villain's Range with Example Calculations

Suppose that in some preflop scenario, we hold A3s and Villain's range is {QQ+, AK}. That gives him 12 possible combinations of AK, 6 of QQ, 6 of KK and 3 of AA, for a total of 27, and 33.33% of that range is KK+. If instead, we held JJ, then Villain would have 16 possible combinations of AK, 6 of QQ, 6 of KK and 6 of AA, for a total of 34, and 35.3% of that range would be KK+. The point is that blockers have the potential to weaken our opponents' ranges significantly.

For the fun part (an example calculation for understanding blockers) I'm going to look at a fun 3-bet bluffing scenario:

Suppose with 100bb stacks, Villain 4x opens a range of {22+, AQ+} from early position. If we assume that Villain only continues against a 3-bet with {JJ+, AK}, then how often does Villain fold if we hold 87s? A4s?

If we hold 87s, then there are 104 hands combinations that make up {22+, AQ+}. Villain continues with {JJ+, AK}, which is 40 combinations. This means that Villain continues 38.5% of the time, and that Villain folds 61.5% of the time.

If we hold A4s, then there are now only 96 hand combinations that make up {22+, AQ+} since we're taking away 3 possibilities each for AA and 44, and 4 possibilities each for AQ and AK. Villain continues with {QQ+, AK}, which is now only 27 combinations. This means that when we hold A5s, Villain continues just 28.1% of the time, and folds 71.9% of the time.

If we assume Villain opened for 4x in EP and we 3-bet to 12x in LP, we need Villain to fold 68.6% of the time to break even just from our fold equity. Against the exact same ranges from our Villain, A4s was a guaranteed +EV bluff while 87s was not because of the blocker.

6. Some Additional Applications of Blockers

This is vry helpful Spoon... esp the thought process around using this to inform EV of bet-sizing on bluffs / semi-bluffs. I put together the chart below a while ago based on something I read somewhere, and use it when playing 'cause my memory's for sh!t. The cumulative number was to help me get an idea of combined frequency, generally rank ordered (emph. generally, as I'm sure could have a whole thread on whether this ordering is correct...), that I could compare to a VPIP or PFR stat as a starting range. "Small blind cards" adds the "none of the above" random crap that is most likely to be completed with in the SB vs. limps.

Thought it was relevant and might be helpful.

I fully expected this thread to be filled with intelligent responses within a day. I hold off on the comment I was going to make, which would have consisted of one of the following-
Nice, Very insightful
So when/where is the sticky going up?
etc.

Yet, no. I come back to find that after 75 someodd views. A single reply(sars post, which is what I was talking about when I said Intelligent responses)

One. That tells me one thing. People click, skim, gain nothing, move on. Probably moving on to a quicker shorter thread that they think will Plug whatever leak they happen to be focusing on today. Is it really so hard? Is there an honest belief that getting better consists of skiming the threads of FTR looking for quick fix solutions? How about take a little time to learn the fundamentals?

BTW:
NH Spoon. This is undoubtedly "sticky-worthy" material.

I totally agree with you. The one thing i was thinking when i read this post was damn i wish there was a way to save to a favorite box of your favorite posts so i could go back over this with out trying to search through all the other posts. I read the whole post and I pretty much understand everything that was said however it is going to take me some practice and several readings to be able to do this at the table. I have been looking for this answer to my question for about 6 months and even posted the question on here several times to only get advice on ev and pot odds and outs. I guesse not knowing the answer I did not know how to properly ask the question.

SPOON THIS WAS WITH OUT A DOUBT A GREAT POST. THANKS

Don

Ship it lazy people holla imo.

Great post.
I personally find it easier to think about the probability in percentage of the deck and not of all card combinations. For example AA would be 4/50x3/49 - and if you did a couple of those calculations, you can estimate the number pretty exactly. For practical purposes you would round it down - 4/50 is about 1 in 12 - 3/49 is about 1 in 15 - 15x12 is 10x15+2x15 is 180. So it's about 1 in 180. I'm sure most people can do that without pen and paper in a reasonable amount of time.

So your example if the board is KKQ - say you hold 63.
Then the possibility of a single person having AA is:
4/47x3/46
or or KQ is:
ok now you got me... I mean there's nothing wrong with doing an approximation (5/47x2,5/46)... but the calculation isn't rocket science either.
1-[(1-3/47x2/46)x(1-2/47x3/46)]
This is first the probability of getting QK and then KQ, and the rest is the "or" part. Actually the probabilities are the same whether you draw the K or Q first, you can cut it short to
1-(1-(3/47x2/46)²)
It's only that complex when there's not an even chance for both of the cars. Explanation is on the first or second page of any elementary probability book of your choice. There's about a 60% chance I got something terribly wrong here... it's been a long time

I missed this until now, hmmm. Great post. It may need to be one of those re-reads for some people after a bit. Like TOP or Gbucks, You may get more every time you read it.

Spoon, you certainly have a way with the teachin'. 6310 is gold.

nice post dude. Im not much of a math guy and feel sorta useless not having much to add, but I just wanted to say that this will probably boost my growth as a poker player to be able to think about ranges a little more deeply. Appreciate you sharing this.

Ok I have read this post about ten times now, so i think I am ready to ask a question. I understan how to calculate using the formula that spoon gives. He is right pretty easy, my question is how to use it.

1. Does this just give you fold equity,
2. How do you calculate how much fold equity you need to break even or gain positive ev
3. How do you figure out that ak is 60/40 dog to qq.

These are pretty much all the questions that come to mind after reading your post. I hope my questions don't sound too broad to be answered.

Thanks for all the help anyone can give me on this.

Don

Final Year Project?
For Your Pleasure?

I know about Pokerstove, I just think it would be good for some people to actually understand what's going on. The math involved in poker is so ridiculously simple... its really something you should understand at some point.

Also you might want to know what the odds are of an overcard hitting the board if you have JJ... what are the odds of an overcard hitting if your opponent has AK, and so on. There are quite a number of things you can't do with pokerstove.

why isn't this a sticky?

I have to admit that I'm totally lost reading this. Which is a shame since all of you have put so much work into it. I have read it a couple of times.

this is brilliant stuff... now here's my question... what is this advanced concept doing in the beginners circle???

This is not tough math however it is hard for me to apply it at the table. I agree as a noob that i would miss out completely if no body took the time to tell us about it or took the patience to work with those who did not understand it. Spoon in no way has left any one out there with info that he is not willing to work with people on. By the way I am still reading this about twice a day until i can put it at the table. I now can sit there at the table and figure out the odds that they could have pocket pairs or ak or what ever the part I still can't figure out at the table is fold equity, however i am still practicing.

I choose to be able to do this stuff in my head once i have it down than maybe poker stove or something to make it easier, But i choose to know how the concept works so i can explain it to my friends and others as well,.

Great post spoon

YOU GUY ROCK is all I can say. I can't believe I have found a site where so many people who are doing so well are SO DAMN HELPFUL!
I'm reading and reading and reading. Then I cut, copy, paste and print. So much info and I want it all. It is very unfortionate I'm not anywhere near the skill level or education as are you the contributors. I doubt I'll be able to sit at a table and put this all to use but I am sure going to try and get my arms around it. I'll no doubt utilize some automated tools but atleast by reading and studying I'll better understand what it is pokerstove and that are telling me.

THANKS AGAIN GUYS!

Oh yeah Spoon you recommend "Apple picking and wall pushing?" Same guy I take it???

Brilliance!
Thanks for the great post.

Great post.

I was thinking about it today, and if you've ever played the guitar or snow skied, (those are two things that I do that I think relate to poker in what I'm about to say, but I'm sure there are more), you get started on a path that will take you from beginner to intermediate, but then you'll hit a wall.

From what I can tell, there are really two paths on your way to becoming great at poker. The first path is from beginner to intermediate, and it's basically just repetition, learning the basics, and more or less just improving upon what you already know. But eventually, you stop improving.

And that's when you need to approach the game from a different angle. You need to start thinking about your opponents cards, what your opponents thinks about your cards, etc.

My point is, the path to being great isn't a straight line from beginner to pro with a stop in the middle at intermediate. It's two separate lines, and if you're not willing to essentially relearn the game from a different angle, then you can't expect to get anywhere significant in the game.

This is another great post from spoon. The concepts in this post will improve my game to another level.

Is spoony still around? Or still on a self ban?

Bumpage

Can this please be a sticky already??

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How-To: Calculating Hand Combinations, About Blockers, etc.

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How-To: Calculating Hand Combinations, About Blockers, etc.