Mathematics in poker by KillerFishes - II. Range - myth called "NIT"
♠_cool: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
