thanks, I'll try 
unfortunately, I slowed down a bit as got sick.
Probability theory and basic hands analysis. PokerStrategy Equilab.
Probability theory introduction.
Basic hand analysis can be found in Harrington's MTT books, in vol.2 if I remember correctly. Enough material about basic concepts like expected values and probabilities ban be read in PokerStrategy articles for various disciplines. For more advanced and better coverage the one can check out “The Math of Hold'em, 2011 by Collin Moshman”.
Let's recap some basic concepts. First of all, it is EV or expected value, the central concept from the Probability Theory ( http://en.wikipedia.org/wiki/Probability_theory ). It won't hurt to read some materials about Probability Theory, especially Discrete case which covers basic poker. Roughly, for discrete case, EV = sum of expected values for all sub-events * probabilities of those events. Sub-events should not overlap with each other and should cover the whole situation we are analyzing. Another essential restriction is that each probability value should be within [0;1] and sum of all considered sub-event probabilities should be exactly 1.
It can be rewritten as EV = ev1 * p1 + ev2 * p2 + … + evn * pn, where p1 + … + pn =1, 0<=pi<=1, i=1..n. In general, EV is sometimes replaces with eq (equity), or even M (shouldn't be confused with Harrington's M-ration).
For example, coin toss, with 49.5% probability of tail when we win $1, 49.5% probability of head when opponent wins $1 and we lose it, and 1% that coin going to stay of its edge and we all pay $10 to the host.
Our EV for each coin toss in $ = 0.495*(+1$) + 0.495* (-1$) + 0.01*(-10$) = -0.1$. i.e. we are expected to lose 10 cent for each attempt.
There are couple of important theorems regarding EV, such as Bayes' Theorem ( http://en.wikipedia.org/wiki/Bayes%27_theorem ) - they are out of scope for now. After you get acquainted with Probability Theory, you can easily derive all those standard poker math formulas mentioned in articles,so you won't need to memorize them anymore and will have much, much better understanding what are you calculating and why.
Example situation analysis. PokerStrategy Equilab usage.
Here is an example hand:
Spoiler
PokerStars - $8+$0.80|125/250 NL (9 max) - Holdem - 9 players
Hand converted by PokerTracker 3
SB: 5,576.00
Hero (BB): 7,243.00
UTG: 6,235.00
UTG+1: 1,561.00
MP: 2,675.00
MP+1: 10,079.00
LP: 5,848.00
CO: 6,172.00
BTN: 6,491.00
SB posts ante 30.00, Hero posts ante 30.00, UTG posts ante 30.00, UTG+1 posts ante 30.00, MP posts ante 30.00, MP+1 posts ante 30.00, LP posts ante 30.00, CO posts ante 30.00, BTN posts ante 30.00, SB posts SB 125.00, Hero posts BB 250.00
Pre Flop: (645.00) Hero has Q♦ K♦
UTG raises to 595.00, fold, fold, fold, fold, fold, fold, fold, Hero calls 345.00
Flop: (1585.00, 2 players) 3♥ 6♠ K♠
Hero checks, UTG bets 775.00, Hero ...
We want to see how good is check-raise on flop.
Ks 6s 3h flop, effective stacks on flop 5610 chips. Pot after preflop 1585 and opp bets 775.
This is a middle stage of the MTT tournament so we assume we can drop ICM considerations and simply calculate EV of actions in chips. PokerStrategy Equilab is used for equity and combinations calculations http://www.pokerstrategy.com/software/10/
We know the villain is straightforward semi-reg and can easily bet-fold with M=10 both on preflop and flop. Based on stats and reads, we assume he opens 17% on the preflop. We assume 100% contbet from him on such flop.
1) Let's consider check-raise all-in move on flop. PokerStategy Equilab usage is described here also.
Spoiler
EV of this move in chips = EV chips outcome for us if opp folds * probability that opp folds + EV chips outcome for us if opp calls * probability that opp calls = (1585 + 775)*Pfold + ((5610 + 5610 + 1585) * probability we win if called + (-5610) * probability we loss if called + (1585/2) * probability that pot is splitted) * Pcall = 2360 * Pfold + (12805*Pwin if called – 5610*Ploss if called + 792.5*Pwe split if called) * Pcall
Now we need to calculate all these probabilities P somehow. Here everything depends on opponent's hand ranges for each action. Let's take the worse case for us, and he'll call us tight with made hand or on very strong draw by pot odds (like strong draw + pair or combo-draw).
First of all, range of preflop open bet: 17% 22+, A2s+, ATo+, 54s-KQs, KQo, KJs.
You can see that this is 226 hand combinations in Equilab, without consideration of blockers in our hand and blockers on flop community cards. They are calculated as: pocket pairs 6 combinations each, non-pair regardless suite is 4*4=16 combinations each, suited non-paired 4 combinations, offsuite non-paired 12 combinations.
22-AA: 6(combinations each)*13(quantity)=78 combinations
A2s-AKs: 4*12=48
ATo-AKo: 12*4=48
54s-KQs: 4*9=36
KQo: 12*1=12
KJs: 4*1=4
Total combinations regardless blockers: 78+48+48+36+12+4=226
So, let's look to blockers now: Kd Ks Qd 6s 3h.
It restricts our opponents to have (you can verify in Equilab if go to suite selection and manually disable mentioned blockers from hand combinations – but it's really time consuming, better read some stuff about basic math Combinatorics):
22,44,55,77-JJ,AA: 6*9=54
33,66,QQ: 3*3=9
KK: 1
A2s,A4s,A5s,A7s-AJs: 4*8=32
A3s,A6s,AQs: 3*3=9
AKs: 2*1=2
ATo,AJo: 12*2=24
AQo: 9*1=9
AKo: 6*1=6
54s,78s,89s,T9s,JTs: 4*5=20
56s,67s,QJs: 3*3=9
KhQh,KcQc: 2
KQo: 4
KJs: 2
Actual total hand combinations for opp on flop: 54+9+1+32+9+2+24+9+6+20+9+2+4+2=183 or 100% case what we assume he has now.
I showed how to consider blockers in manual calculation, but Equilab makes life much easier: you can select exact flop board and your hand, then click on hand range pop-up for the villain and select that total range again, now it with highlighted checkbox on it takes into account mentioned blockers automatically:
Range of call: pocket pairs JJ-AA (we include some bluff catchers, opponent need to take some reasonable risk with such vulnerable stack and our check-raise all in might be semi-bluff); combo-draw 54s of spades, A3s of spades; top pairs AK,KQ,KJs; sets 33,66 – this is (manual calculation): (6+6+3+1)+1+1+(2+6)+(2+4)+2+3+3=40 combinations, or 40/183=0.218579235, about 21.86% probability.
Again, you can calculate these 40 combinations in Equilab in the similar way, the only complication now, that you need two particular hands As3s, 5s4s to be selected: select the rest range as normal, then click “Suite Selection” button, select A3s,54s and click “Suite Selection” button again, it will open one more pop-up where you can select required suits and then close, as a result you get 40 combinations:
Therefore, opp folds in the rest of 78.14%.
Let's look to Equilab how good are our KQs in case of call for the selected range: 32.01% victory, (7.54+7.54=15.08)% split and 52.91 % loss with total of 32.01+7.54+7.54+52.91=100
Therefore, EV of check raise for us on flop in chips = 2360 * Pfold + (12805*Pwin if called – 5610*Ploss if called + 792.5*Pwe split if called) * Pcall = 2360*0.7814+(12805*0.3201-5610*0.5291+792.5*0.1508)*0.2186 = +2117 chips.
Sure, the formula for EV can be much shorter here if you use Equities from Equilab, but it's a derived formula. I wanted to demonstrate original formula from Probability Theory. Equity formula will be derived in future posts.
2) Small check-raise on flop.
Spoiler
I won't go into details here, too complicated to analyze as opp can call in position with draw and need to calculate turn scenarios. Plus with check-raise all-in on wet flop we balance our strong draws.
In case of dry flop the situation would be completely different, there won't be much scare turn cards for our stack size (except A), and no need to balance draws and be afraid of opponent's draws. Thus, small check-raise may be a better solution.
3) Lets return back to preflop and consider our resteal-push.
Spoiler
We are risking 5955 chips. This opp will call us with 88+,AQ+.
Now we got only 2 blockers: Kd, Qd, you can specify them in Equilab as a specific hand for one of the player and calculate total combinations and call range combinations on rang pop-up as was described previously.
Total opponent's combinations: 208 or 100%
3bet AI call range: 60 combinations or 60/208=0.288461538, 28.85%
Therefore, fold probability is 71.15%
Preflop AI of our KdQd vs selected call range in Equilab is:
34.78% victory, 64.82% loss, (0.39+0.39)=0.78% split.
EV of our 3bet push preflop in chips = 1240*0.7115 + 0.2885*EV when we are called = 1240*0.7115+0.2885*(12805*0.3478-5955*0.6482+0.0078*645/2) = +1054 chips.
It doesn't necessarily mean that it's better to call KdQd instead of push, as we hasn't analyzed the whole line with call preflop – it should consider those calls on BB when we check-folded without hit, and check-pushed with OESD and flash draw, or combo-draws, or Qxx boards.
