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Mathematics in poker - I. 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! )

P.S I moved it from Self Study... I guess this is better place
• 60 replies
• Bronze
Joined: 16.04.2011
Never been good at math, but perhaps i should study for this game, one question, you seem like a math guy what is the difference between what you know about math and what knows apro like Chris Ferguson, they say he is a pro that knows alot of maths, i mean what is more to poker in the math section besides odds, pot odds folding equity, and implied odds?
• Bronze
Joined: 16.04.2011
As far I could say, and correct me if i m wrong, poker is more about psychology, and yes maths but basic math hand ranges, odds not to complex mathematics
• Bronze
Joined: 11.07.2010
I recommend you to study for the game, because otherwise it will be very hard for you too reach good winnings.

Chris and a lot of others belongs to group, which is "math based". This group of players is usually made from older conservative players. They rely on math a lot. On the other hand there's group of players as Durrr, Ivey, Gus ... These players are more based on making presure on their opponets by doing big bluffs and bets. However, they all MUST know about odds, implied odds, other way they would lose from the very start.

About the second question...
Poker is IMHO about making +EV moves.

Whether I call a raise or bet, I MUST do it +EV and you can't know, what is it if you can't make math ...

EV = P*DPo
P... probability
DPo ... decimal pot odds

I started from wide so you have some basic informations and we can use it further in next topics
• Bronze
Joined: 16.04.2011
Ok so they are specialised in math, but this stuff that you show me i can find on the internet, more or less, but what kind of math do these guys know that makes the difference between you and them i mean between you a mediocre player and them pro players
• Bronze
Joined: 16.04.2011
Hey search GAME THEORY in poker you ll be surprised about the psy aspect of the game
• Bronze
Joined: 01.04.2009
Originally posted by KillerFishes
It means we can get 1326 combinations of our 2 cards. Pretty cool right? This number we will indicate as "m"
Knowing these are not unique combinations is worth more than knowing that there are 1326 total combinations.
• Black
Joined: 06.12.2009
Originally posted by tokyoaces
Originally posted by KillerFishes
It means we can get 1326 combinations of our 2 cards. Pretty cool right? This number we will indicate as "m"
Knowing these are not unique combinations is worth more than knowing that there are 1326 total combinations.
if you want to do proper equity analysis with just pen and paper, it actually isn't.
• Bronze
Joined: 01.04.2009
Originally posted by Shevtshenko
if you want to do proper equity analysis with just pen and paper, it actually isn't.
There's a reason that Sklasnky isn't rich from playing poker. It's because mathematicians suck at playing poker.

I do look forward to your ebook though.
• Bronze
Joined: 22.11.2009
Originally posted by tokyoaces
Originally posted by Shevtshenko
if you want to do proper equity analysis with just pen and paper, it actually isn't.
There's a reason that Sklasnky isn't rich from playing poker. It's because mathematicians suck at playing poker.

I do look forward to your ebook though.
• Black
Joined: 06.12.2009
Originally posted by tokyoaces
Originally posted by Shevtshenko
if you want to do proper equity analysis with just pen and paper, it actually isn't.
There's a reason that Sklasnky isn't rich from playing poker. It's because mathematicians suck at playing poker.

I do look forward to your ebook though.
There's a lot of things in poker that are possible to be solved using math. Some of those have been solved and if you don't know about them and don't want to know about them, that's not my problem. I don't want to judge you... but saying that being good/very good in math does not help your poker game is just wrong. Equity calc's, range calc's, game theory, logical thinking etc. all those are based on math and are in the essence of poker.
• Bronze
Joined: 14.05.2011
WOT A LOAD OF RUBBISH. STOP OVERCOMPLICATING THINGS TRYING TO SOUND LIKE YOUR A PRO AT THIS.... ONLINE POKER IS 80% LUCK OF THE DRAW AND 20% SKILL IN HOW U PLAY UR HAND... STOP WITH THE WHOLE MATHS IMPORTANT STUFF.
• Bronze
Joined: 14.05.2011
U R RIGHT ITS FAR MORE PSYCOLIGICALLY BASED THAN MATH. U SEE PLAYERS LAY DOWN TOP PAIR WHEN THEY ARE RAISED WITH AN AK ITS ABOUT KNOWING/THINKING YOUR WON/BEAT END OF.
• Bronze
Joined: 14.05.2011
WOW U REALLY OVERCOMPLICATE THINGS......
• Bronze
Joined: 22.11.2009
Don't forget game theory.
• Bronze
Joined: 23.06.2008
...and the CAPS lock!
• Bronze
Joined: 11.07.2010
Well I didn't mean to sound like
"If you don't know this, you can't play good poker" ... It's not true ofc ... Math is just one aspect of poker ... It helps you to make your decisions so I've guessed it will help you to improve

Sorry if somebody is irritated ...
• Bronze
Joined: 18.01.2008
I don't see how knowing more than pot odds, implied odds and ranges is useful to poker. I'd be glad if you proved I'm wrong though
• Bronze
Joined: 30.11.2007
It's because mathematicians suck at playing poker.

Dear jbpatzer ,
I think it's not applicable to u cos I have many doubts (from ur blog,nice blog btw) that ur not true mathematician.
• Bronze
Joined: 07.08.2009
Originally posted by Tosh5457
I don't see how knowing more than pot odds, implied odds and ranges is useful to poker. I'd be glad if you proved I'm wrong though
just because im bored ill try to give u small example

u raise with KQ on BU and BB 3bet. His 3bet range BB vs BU is 15%, his fold to 4bet is 60%. Given this numbers you can calculate if it is +EV to 4bet and fold with KQ BUT in fact you calculation wont be correct. JUST because you are holding K and Q the COMBINATIONS of good hands in his range decreases greatly (yes you can calculate this with exact percentage) therefore it is much more likely for him to fold to your 4bet. You can do this exact calculation only if you know how to count hand combinations.

of course then come things like meta game, history, reads etc but this is other topic