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# Mathematics in poker - Combinatorics - Basic of Basics!

• Bronze
Joined: 11.07.2010
Hello guys,

I started new thread called "Mathematics in poker" obviously it will be about Math in poker ... I enjoy counting everything and a lot of it I found very interesting and I surely used it in all formats of Poker (CG, SNG, MTTs doesn't matter)...

I want to start with basics of Combinatorics (I think everybody has some experience with that, however, if you don't it's ok. This thread is wrote step by step!)

How many combinations of 2 cards we can get? And using this knowledge in poker. It's worth of milions \$!

A lot of people skip this with "It's so useless"... They think this isn't poker, they think this can't improve your game. They're wrong and I will show you how powerful it can be

At first, we need amount of all combinations of 2 cards because with this we can calculate all other probabilites based on this formula:

P(A) = means probability of our phenomenom
m(A) = positive phenomenoms
m = all phenomenoms

We can use this formula only if we can consider that all phenomenoms have same probability (means probability of getting AA and 22 are same)

Now, we need some more formulas, how we can count the amounts of combinations? We always get 2 cards from a deck - 52 cards. For being able to count that we need some more formulas - I will explain below.

The first is called Combination I indicate it as K.
n ... all elements (52 cards for us)
k ... means how many elements do we take for NL k=2 (we take 2 cards from 52), for omaha it's 4 (we take 4 cards from 52 cards)
n! ... factorial = for example 5!= 5*4*3*2*1 ... 7!=7*6*5*4*3*2*1 I hope you get the idea

Now, based on all informations I gave you, you can try calculate how many combinations we have in poker - it means how many different combinations you can get.
(TIP: We take 2 cards from 52 cards)

It means we can get 1326 combinations of our 2 cards. Pretty cool right? This number we will indicate as "m"

We are slowly getting into using of these knowledge to poker, these are still basics, however, be sure that a lot of people have NO idea about this and it means you can have a big edge

Ok now .. How many combinations do we have for AA and AKo + AKs? Try it on your own!

For AA:

It means we have 6 combinations for 4 cards. Now the basic of basics - count the probability of getting AA. It's easy just use the very first formula.

P(A) = m(A)/m =6/1326 => ~0,45%

AKs + AKo:

I hope you discovered why we have to deduct 12. If not think one more time. We are doing combinations of 8 cards, however, we don't want to count paired (Aces have 6 combinations paired - Kings the same = 12) that's why we have to deduct this amount to get number 16.

There are 16 combinations of AK. How many of them are suited and unsuited? It's not hard to figure out. We have 4 colours so it means:
AKo = 12
AKs = 4

As you can see, there is higher probability of getting AA than AKs!!
So what are the probabilities of getting these cards?
Phenomenom A ... Getting AKo
Phenomenom B ... Getting AKs
Phenomenom A u B ... AKo or AKs
Again the first formula:

P(A) = m(A)/m = 12/1326 = 0,9% for getting AKo
P(B) = m(B)/m = 4/1326 = 0,3% for getting AKs
P(A u B) = m(AuB)/m = 16/1326 = 1,2% for getting AKo,s

Ok ... I think this is enough for now, the basics of all basics are behind us ) Now you can try to count a lot of different situations. You MUST be really confident in these numbers, next time we will take a look at some basic stuff as hitting flop, turn, river and opponents ranges.

If you have any questions, just write it here I will answer as fast as possible and I hope we will be able to figure the problem out

Have a nice day! )
• 11 replies
• Bronze
Joined: 22.11.2009
• Bronze
Joined: 11.07.2010
Originally posted by jbpatzer
The 3bet/4bet/5bet/shove game

Alternatives to ICM?
It looks like I'm wasting my time here, you did a great job!

However, I think I will have a little different approach don't you mind if I will continue? I want to give something back to this community Even though, your threads looks great and you are probably way ahead (I'm only on highschool so you should be ))
• Bronze
Joined: 22.11.2009
I wasn't suggesting you should stop, just adding my input. Keep at it!
• Bronze
Joined: 11.07.2010
Jbpatzer: Hehe ok, thank you

Let's see if people will like it and find it worth reading.
• Bronze
Joined: 25.03.2011
I approve of both of your posts,i love combinatorics and will do a in-depth analyses of your thread and jbpatzer's thread.

Good job.
• Bronze
Joined: 03.08.2009
there was once a nice blog looking at variance - cant remember who's it was tho
• Black
Joined: 02.11.2008
As the girl from Jerry McGuire would have said, "you lost me at 'Hello'"...

• Bronze
Joined: 08.10.2009
nice
• Bronze
Joined: 11.07.2010
Thanks I will add more articles, however, I'm giving priority to new Czech forum There are already 3 series ...

Plus it's quite hard to express myself in english in order to not write nonsense ) However, I will try to work on it ... Next will be ranges and reading stats, you can try to write here your tips of range PFR 10%:

without programs!! ) I will give you a piece of advice how count it quickly
• Bronze
Joined: 31.12.2013
Good work, finally a nice explanation on how the mathematics behind it work. I am no math guy but i clearly understood it after reading it carefully.

It is refreshing to see the math behind it all, instead of posts just blurting out oh for pairs you have 6 combos and for 2 other cards like AK you have 16 combos etc...

Thanks again and kudos for doing this.

I hope in the future i will learn to use this information to my advantage.
• Bronze
Joined: 27.09.2009
One helluva bump!