Fold equity and Float equity

I once heard Antonio say,"Poker game theory dictates that no one ever makes a hand. So just bet and win!"  While I understand the subtle truth behind his sarcasm I wanted to quantify it. I took a reasonable range of hands for a Tag  and a Lag in holdem' and actually calculated the percent of times that they hit these flops with hands that they would want to continue with (straight draws included). This is a rough guideline and it ignores flush draws but I believe it is a good reference.  The percentages represent the "steal ability" of the hand. For instance The Lag will miss and probably fold to a 3 bet on a 552 board 78% of the time.

         

FLOP                    TAG                   LAG

A72                        47  %                  66%
K72                        59 %                   63%
Q72                        51 %                   67%
J72                          61 %                   53%
T72                         73  %                  61%
552                          73  %                 78%
JT3                          48  %                 29%
QQ3                        57  %                 71%
This chart does however take into account card combo odds. For example while pocket jacks are in the Tag's raising range and there are 6 different combinations of jj ♠♣, ♠♥, ♠♦,  ♣♥, ♣♦, ♥♦. However if there is a jack on the board there are only three combinations possible. This is reflected in the percentages.

So what to do with this information? Its obvious that people do not hit low boards or paired boards often. So does this mean that you can steal often? I believe with good taste you can carve out money in some situations.
NL HOLDEM 1-2$
Situation 1:  classic
A couple of limpers and you limp in with jt♦, the button a standard LAG
raises to 11$ the two limpers call and you call pot is now 47$
flop is 552 rainbow
Action checks to the button who bets a standard 30$
The two limpers fold.
pot is 77$
(Of course this scenario requires that you have a decent image and enough effective stacks to make a bluff plausible, in this scenario well say that the villain has 250$ and you have him covered.)
So there is 77$ in the pot and we know that he is likely to fold to a 3 bet 78% of the time. or (3 times out of 4) or 3-1 so if you raised you would have to raise at least 30$ more meaning that you would put in 60$ I believe to raise here I would probably make it about 80$. so I'm risking 80 to make 77or about 1-1.
Obviously profitable.  3 times he folds for +77$ or 3*77=231
                                  1 time he calls for - 80$   or 1*-80=-80
                                                                231-80= 151$(in 4 hands)so 151 divided by 4=+37.75 per hand (or per situation)
This situation is very lucrative according to the math. I understand that this doesn't account for any other players  or re-bluffs or other intangibles. I would like to think that these pitfalls are cancelled by the value created on future hands. Such 3 betting "light" is easily observed by the table and if you are looked up your image has become that of an "action" player. With this in mind lets say that there is one more player behind you and he limp-called pre-flop so his range is something like 22-99, qj-qt, jt-j9,kj-k9 a9-a2suited (288 combinations) and 22 55 and A5 are the hands that he will call your 3 bet with or (16 combinations) that's about 6% of his range. So the equity of this hand has gone down from 78% to about 72%. With out laying out the math its easy to recognize that this does not make the play negative expectation. It should be noted that a 3-bet into 2 players is more credible than a "heads up" play.
Also if you are in position you can float one bet and bet 40 on the turn. Or maybe you feel strongly enough about the situation that you can float out of position. The math generally stays similar except for the chance that  he will hit one of his overs (13%). But in reality you don't like any card higher than  a jack ( you really do not like a jack because the type of hand we are saying he is most likely to have is one that frequently has you dominated) so the  actual odds of an unfavorable card is 15/47 or 32-15 or 32% or about 2-1. So the float play (if you could say for sure that if you flat-call the flop then 90% of the time he folds the turn if no scare card comes is 90% of 68% of 78% or about 48% equity. so about 1-1 which means that flat calling the flop should be about even money. Which once again I believe is still positive ev because of the implications on future hands.
     I would like to make clear that this is a simplistic idea. It was only meant to give an idea of how comfortable you should feel in these situations. If you had a hand like AK in this spot you might (correctly)  feel like you are value betting instead of bluffing.

Realistic expectations in a tournament

 I recently listened to an interview with Negreneau who said "in tournaments you should generally not be looking to protect your hand or build big pots but to avoid going broke".  He made an example of a hand where you raise it up with two jacks and the big blind calls and the flop comes q56 with two diamonds and you bet and he calls. now the turn is a brick and he checks. Daniel recommends checking here every time just to avoid the big pot and going broke. ( Don't really understand why the queen was on the board in this example) That's great advice for deep stack tournaments but a lot of tournament situations consist of being 30 big blinds deep or less.
Thinking about this I began to wonder what the odds are of being all in with an over pair 4 times and winning every time. I believe that its basically 80% four times. which comes out to about 41% How about dodging the flush when your opponents flop 4 to a flush 4 times in a row. 18%. or actually flopping a set versus an over pair and winning 4 times in a row with your set.60 %.
 So to win every time set versus over pair 4 times in a row happens only 60% of the time!  That's a very low number to me  and how often does that happen in a tournament anyway?
So lets say you have run deep in a tournament and along the way you avoided a flush draw 4 times. Had a set versus an over pair 2 times. Had an over pair versus a under pair 3 times.  and had someone out kicked 4 times the odds of winning all of these is only 4%. So in this tournament it could be said that you just ran like god. And  that's my advice kids "run like god".

Two kings and my third nut.

Two kings and my third nut.

1-2 NL
My hand KK in the BB
Limps to me in the bb so now were 4 handed. I felt like  limping here  because I have been very active and i felt like I have created a lot of value for myself in pair vs. pair situations . For example I don't think anyone was wanting to fold kj to me on  a j52 board. I almost always raise in this spot though.

So I check my big blind.
Flop is K Q T rainbow
Action is: check,I check, bet 10$,button flat calls 10$,

fold, I raise to 50$, original raiser folds,button raises to 110  By grabbing a barrel and just slamming it down out there inside the betting ring. A gesture of which "q" a great player would say is strength. In his words," a come and get some bet".

So now the pot is 188$ and its 60 for me to call. 3-1. The button has 211 more and i have him covered. I felt like I was beat like 85% of the time. Sammy, for those who don't know him you better ask somebody, stated that this 3-bet is always the nuts. His opinion is that everyone at this level will flat call or fold with everything else.

So its 3-1 to me. The odds of me hitting my full house on the next card are 5.7 - 1. So to break even I need to make 5.7 times the 60$ or 392$. That's 204 more than whats in the pot now. So essentially i need to get his whole stack once i hit  the full house. When I do hit my hand the board will be paired and I gotta believe I will stack off most players regardless, but some people will get away from a straight. So the small percentage of the time a player folds the turn or river because the board pairs it makes the implied odds incorrect. So this play is wrong to me.
Also if I call and miss the turn. The pot will be 248$ The action will always be something like check, bet . I think he will bet something like 100 - all in. So if he bets 100 I will have to call getting about 3.5-1. and hitting the full house here is 3.7-1. But really he will probably stack off regardless of what comes. Now that he only has 111 left and a 448$ pot out there. So really I'm getting about 4.6- 1 . So i will have to call. If he pushes all in on the turn I will have to call 211$ to make 459$ or  2.2- 1 which is incorrect.
 So calling the flop is wrong. Folding the flop seems weak, but correct right now. Pushing is the last option.

Pushing all in:
My equity is 35%( odds of making my full house)
my fold equity is 5% or less.( could be 0% I believe he could have j9 a small percent of the time. I also have seen strange things like people folding the nuts!He could be bluffing I could see someone just getting sick of me!)
So 5% of the time i win 188$    or                     +9$
35% of the time I win    610$    or                     +214$
60%  of the time i lose   211$    or                      -127$
for a total ev of                                                       +96

It was a tough and weird spot. The pot was limped and so it was tough for me to put him on Aj. I also had thoughts like he cant give me credit for a real big hand because it was limped. Obviously hes not afraid of j9 also with the re raise he put in.

In my head the math was simpler. To call I need about 6 times the 60 or 360 or about 200 more than whats in the pot. Break even at best. To push Ill win 1/3 of the time or 1/3 of 610
or 203 I thought at that number that it was right because he only had about 200 left. and the rest of the equation  could only add up to me losing a percentage of my remaining effective stack.  Also at the time i thought i had like 15% fold equity. Now I believe  ,in that spot, its a lot less. 

So If I push the Ev is +153, and If I fold I keep my 211 so the decision, from a hard math standpoint,  should have been to fold.

Hand ranges and drawing hands.

In the theory of poker sklansky advocates betting an amount that makes it incorrect to draw and is less than what they will fold for. That is if you have showdown value.
So when the pot is 10$ and they have a flush draw you need to bet at least 3.13$ seems ridiculous right? The equation is
pot+bet=4.2*bet
or
10+x=4.2x
because 4.2 is the odds against hitting your flush on the turn. which simplified means reducing 4.2x to 3.2x=10 or 10/3.2.
So simply  reducing the draw odds down to 1 less is an easy way to find the break even point. I know this seems real mechanical and ridiculous because we all bet 2/3 pot or more and they always call. And how do you bet 3.13$?
I have a point, wait for it.
When you think they have a straight draw the bet should be at least 2.56$
These numbers are extremely low. Lets turn the situation around. My point is how often should you call the warmers?
Say you have a flush draw and you check to the warmer and he bets 10$ and you want to call. If you call to hit your flush you need to make 42$ There is 20$ in there already so he needs to call or bet 22$ more either on the turn or river.Does this happen a lot? I don't think so not when there are 3 of a suit up there. I think the action a lot of times goes check check turn then bet fold the river. Or maybe he calls 15$ on the river, maybe. I like to lead the turn because its a weird line and with 3 to a suit he could have one of them and decide to draw. The real value comes from other players in the hand. The implied odds of them hitting a hand like a flush also. So the idea is to have big flushes in multi way pots right? i think I drip a lot of money always assuming there will be other callers making my odds correct. Or that I will make enough money off of the player when i get there. I also know that raising is a better play. Check raising is a strong play that sends the message that a lot of money is going in on the turn and the river. So the tags  and taps could fold a lot of hands.
Full ring 1-2 NL
All players are 150 deep
4 Limpers including you with Ah4h
On a kh7h2c board you check and the average player in position bets 10$ into a 10$ pot
I think his range is something like
22   3 possible combinations of this hand                                     3%
77   3 combos                                                                                 3%
kq   12 combos                                                                               15%
kj    12 combos                                                                                15%
kt    12                                                                                             15%
k9    12                                                                                            15%
k8s  3                                                                                               3%
k7s  3                                                                                              3%
tt     6    not likely but in this hand he was checked to in position  7%
aa    6    people do like to get tricky sometimes!                            7%
kk    3                                                                                                 3%
56h  1                                                                                                 1%
89h  1                                                                                                 1%
45h  1                                                                                                 1%
68h  1                                                                                                 1%
9t h  1                                                                                                 1%
tj  h  1                                                                                                 1%
qjh    1                                                                                                1%

Skip down to the ♣ if you want to skip this boring section.

If your image is decent and you raise he will probably fold most of  these hands: all flush draws 7%, all one pair kings 51%  tt3% so ur fold equity is around 61% He flats you with AA 22  kk and k7 or he flats you 17% and he re raises you 22% and  I know that the card combination odds should be weighted toward kx type hands and maybe a little less than 3% for the 22 and 77 because alot of players would rather check with these hands but it only changes the math slightly. Also some players will re raise with 22 and 77 every time but some do not. A lot of intangibles could be figured and will change this math slightly but make it very cumbersome and hard to read.

If your image is a little crazy then maybe they only fold king x  half of the time and decide to draw against you half of the time. So your fold equity drops to 37% He flats you 34% and re raises you 29%

So the action is to you kh7h2c.
Your image is decent and you decide to check raise to 40 .
So now 61 % of the time he folds and you win the 20 or +12$
He re raises you off of the hand 22% for -9$
17 % of the time he flats you and you hit your card on the turn 19% of the time.or about 3% + implied odds of future streets.
17 % of the time he flats and you miss or about 14%

Implied odds: He calls you hit the flush.
 Now if he flats and you hit your card I think given the action of this hand that he will call at least one more bet  50% of the time whether you check the turn bet the river or you bet the turn then he folds the river( not likely)  lets say you bet 60 so that's 45% of 60  or + 27$
 35% you can get it all. or 35% of his remaining 108$ + 38$
17% he folds + 0
     3% o f the time he full houses and gets all of your stack  - 3$
so the implied odds of hitting your card  after he flats you are a total of +62$ + the pot 20$ or 82$
3% of 82 is about 3$

reverse implied odds he flats you miss:
ev situation without laying out the math 30% of the time he pushes in and you have to fold for - 40 or -12$
50 % of the time he checks behind and you hit your card 20% for 10% of  20$ or +2$ or about - 5 $
14% of -5$ is  -.70cents
So if he folds                            +12
He re raises you                        -9
He flats :
      you hit                                +3
      you miss                             -.7
                                            +   5.3$ ev

with a crazy image:
he folds 37%     + 7 $
he raises you off the hand 29% or -11.6
he flats :
he flats you hit 29% of 19% or 5% total
implied odds If you hit your hand its less likely that he has a flush also but it ensures that you get the whole stack. I would guess that with your image you get it all a lot of the time like 65% so + 70.2
10% of the time you only get one more 60$ bet + 6$
22% of the time they put you on the flush and fold +0
3% they house you and get your whole stack -4$
total expectation of 72.2$
5% of 72.2 is  +3.61$

he flats you miss:
40% action goes check bet and you have to fold I believe that since my image is crazy that if I check they will bet here and bet enough.
60% it goes check check so 12% i get there and lets say for simplicities sake that 50% the time i win one more bet or 50% of 60 $ so + 30$ so 12% of 30$ or +3.6$
the remaining 48% of the 60% is when it goes check check  and i pretty much lose the hand. I miss the draw and they are not folding alot of the time on the river for one more bet. so the expectation here is 48% of 40$ or -19.2
so the total expectation of him flatting and you miss the turn is -15.6* the remaining 24% of the he flats you miss category= -3.74
So the total ev of check raising with a crazy image is
he folds                                         +7$
he raises you off the hand               -11.6
he flats:
       you hit                                    +3.61
       you miss                                 -3.74
                                                     -4.73  total ev
♣ So a + 5.3 ev if your image is decent and a -4.73ev  if your image is crazy. I know you will disagree with a lot of the probabilities and percentages i assigned but i do not believe that they will change the conclusion  here.     I know I could treat an ace as an out also but maybe its not. This is  in a game where you are playing a lot of hands and a lot of hands out of position. Its a somewhat typical situation and you can deduce yourself how more players in the pot and how loose your opponent really is can help you or hurt you. It seems to me like you need to have the ability to steal the pot out right with a crazy image and only bet made hands. As a decent imaged player you can maneuver more. Maybe you could make a case for barreling the turn. I believe that once I re raise the flop the hand range is pretty polarized and maybe ill pick up a few more folds on the decent image ev side. So this will only make the conclusion clearer.              

Getting thrown out of the Poker room.