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PL/NL Texas Hold'em >> Micro Stakes

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matrix
there is NO spoon

Reged: 09/04/05
Posts: 7050
Loc: UK
Basic Theory: -Expected Value-
#7853639 - 10/30/06 08:20 AM

edited to fix glaring typo in 2nd example and to clear up AA v KK example - edits are in italics (matrix128)

Expected Value - is commonly referrred to as EV.

from here on in positive Expected Value is +EV and negative Expected Value is -EV.

Poker is a game in which skill will beat luck every time assuming that you play for long enough. While it's true that any two cards preflop can win any given individual hand and that luck is a large part of this game if you hold any aspirations whatsoever to beat Poker overany significant amount of time/hands you must learn to make +EV plays and not make -EV plays.

EV is simply what you expect to make on average with any particular play.

here is a simple example

Hero(100BB) has A A and raises preflop to 4xBB from the CO.
Villain(100BB) calls from the BB and both see a HU flop of 9 3 6

Villain tells us he has black Kings (he's not lying) and then raises all-in and Hero calls.

Villain tables K K

(disregarding how good the play is in this hand) what is the EV of calling knowing we are against specifically K K ?)

If we punch those numbers into Pokerstove we get this output..

Board: 9c 3d 6h

equity (%) win (%) tie (%)
Hand 1: 08.3838 % 08.38% 00.00% { KcKs }
Hand 2: 91.6162 % 91.62% 00.00% { AcAs }

we can see here that if this hand goes to showdown (as it is going to) that Hero will win on average ~92% of the time.

so if we run this hand 100 times Hero ought to expect win 92 times and lose 8 times.

there are ~200BB at stake so Hero wins 18400BB the 92 times his AA holds up - and loses 1600BB the 8 times he loses the hand.

Total net win of 168BB/hand.

This play is +EV and has an EV of 168BB *every* time you make it.

It's important to note that EV and actual results can vary massivley over any short term period. e.g. if we actually ran the hand above 100 times you might win all 100 times - does this mean the EV has changed? or you might be unlucky and lose 25 times in 100 - does this mean the play is now less EV? - no EV remains 168BB per hand. Everytime you make this play you "earn" 168BB and the more times you repeat this the closer your actual real results will get to the "perfect average" of winning 92% of the time.

Once you have played enough hands (an infinite amount) your total actual results will equal the sum of all of the total EV of the plays you have made. The closer your total number of hands gets to infinity the closer your actual results will get to this theoretical figure. So in theory every time you make a -EV play and get chips in when you are an underdog you a "losing money" regardless of the actual results of the hand - and conversely everytime you get chips in when you are a favourite in a hand you are winning money. If you added up all the "Sklansky Bucks" (theoretical EV money) you made in the long run and compared this amount to your actual winrate - after playing an infinite amount of hands these two numers will be identical - and the more hands you play the closer these two numbers will get to each other.

Lets look at a more complicated example, in our simple example above we knew villains exact hand before calling so we don't have to put him on a range (which affects the EV of our play) in practice we never know what particular hand we are against when we make our decisions. This is a real hand from my database.

Poker Stars
No Limit Holdem Ring game
Blinds: \$0.10/\$0.25
6 players

Stack sizes:
UTG: \$27.85
UTG+1: \$24.65
CO: \$28.95
Button: \$23.95
Hero: \$25.15
BB: \$27.80

Pre-flop: (6 players) Hero is SB with 2 2
UTG calls, 2 folds, Button calls, Hero calls, BB checks.

Flop: J 2 5 (\$1, 4 players)
Hero bets \$1, BB raises to \$3, UTG folds, Button calls, Hero raises to \$8, BB raises all-in \$24.9,Button folds, Hero calls.

Turn: 9 (\$53.8, 1 player + 1 all-in - Main pot: \$53.8)

River: 9 (\$53.8, 1 player + 1 all-in - Main pot: \$53.8)

Results:
Final pot: \$53.8

- it's the flop action I am interested in here.

In real life we don't know what sepcific hand we are facing at the point in time where we make a decision. What hand does BB have here? is my hand strong enough to call his all-in? and how do we work out the EV of this play??

The answer is to put BB on a range of hands - if we re-run this hand 1000 times say sometimes he has AA and we are a huge favourite, sometimes he has 55 and we are a huge underdog, he might also have JJ-KK, AJ,KJ,J2,52,J5, Ax, or he might be bluffing. In this particular case his range is wide because there was no preflop raise. Also we are not saying htat he will always play every hand in this range exactly this way - but that he isn't playing any other hand apart from the ones in this range in this fashion.

Against most of these hands I am a favourite, and against some of them I am an underdog. I have no way of knowing what hand he has and certainly don't have time at the table to put the numbers into Pokerstove so we just make an educated guess.

I play using the general rule that I should never fold a flopped set for ~100BB. The reason being that no matter the flop if we can get all the money in on the flop we are almost always a favourite to win the hand at the showdown vs our opponents range of hands.
So I happily call his all-in. But have I made a +EV play and will this earn me money in the long run???

Lets put his range and my hand into pokerstove and see...

Board: Jc 2h 5h

equity (%) win (%) tie (%)
Hand 1: 78.7155 % 78.72% 00.00% { 2d2s }
Hand 2: 21.2845 % 21.28% 00.00% { JJ+, 55, AhKh, AJs, J5s, J2s, Ts7s, 52s, AJo, J5o, J2o, 52o }

(T7ss is included in this range to represent a bluff)

and the numbers say that on this wide range of hands my play is +EV and that calling his all-in here means that vs that range I expect to win ~79% of the time.

The actual results don't matter, as long as my range is accurate, and what cards come on the Turn or on the River don't matter either (as the decision is already made by then) if I make this play everytime it is +EV and in the long run I expect to win ~170BB everytime I make this play. As this play costs me 100BB to make I make a profit everytime here of 70BB, whether BB shows me JJ for top set or A 8 for a busted flush draw I still "gain" ~70BB everytime I make the play.

Whenever you determine at the table that a play is +EV you should make it EVERY time. If you don't you are losing money in the long run. Do You See Why?

Ultimately it is EV that will decide what your true winrate is, you can't beat it, or get around it in the long run eventually your total real results will match your expected results.

Closely tied in with EV is variance - a lot of people misunderstand what variance is and try to avoid it. But you shouldn't. The very very best players at poker don't care about variance and try to make every single +EV play that they can (this is the main reason why they are such big winners) Variance is simply how much your actual results can vary from the statistical EV results in the short term. It's the reason that a 20x buyin roll is recommended. So that you don't go broke in the short term making +EV plays that you lose in the short term because the real results vary from the Expected results. Variance is neither good or bad - and the bigger bankroll you have to absorb variance the more you ought to be willing to risk on a marginal +EV play.

Lets say you determine that a play is +EV and you'll win 51% of the time, the more money you stake on this play the more you stand to win in the long run. 51% of 200BB is more than 51% of 20BB - though in the short term real results will vary lots and you stand a great chance of losing this particular bet if you can afford it (have a large enough bankroll) you should bet as much as you can on this 51% shot.

As a final thought here is an exercise you can try when you next get a big losing session.

Review all the hands in the session and for each hand you play work out a range of hands for each villain, run the numbers into pokerstove and see how much you made in EV.

I do this sometimes and often find out that I had a +EV session that in real results lost me lots of real money. If most of the losiung sessions you have are +EV you are paying well and eventually real results will catch up with your EV results and you will be a long term winner, so despite losing now in the short term you can be happy that in the long run you're still winning

Edited by 4_2_it (10/30/06 01:50 PM)

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uminchu
veteran

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Posts: 1482
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Re: Basic Theory: -Expected Value- [Re: matrix]
#7853706 - 10/30/06 08:38 AM

NICE Post man

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CaptVimes
old hand

Reged: 09/13/06
Posts: 992
Loc: Embracing Distractions
Re: Basic Theory: -Expected Value- [Re: matrix]
#7853710 - 10/30/06 08:40 AM

NH, Thanks for the knowledge.

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munkey
A model citizen

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Re: Basic Theory: -Expected Value- [Re: CaptVimes]
#7854385 - 10/30/06 10:27 AM

bump

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bozlax
Just a big [censored] softie at heart

Reged: 12/29/04
Posts: 8848
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Re: Basic Theory: -Expected Value- [Re: matrix]
#7854653 - 10/30/06 10:55 AM

Quote:

Total net win of 168BB/hand.

This play is +EV and has an EV of 168BB *every* time you make it.

Two out of three of these statements are wrong. Do You See Why? Hint #1: this play is +EV. Hint #2:

Quote:

Villain(100BB) calls from the BB and both see a HU flop...

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pokerchap
Pooh-Bah

Reged: 06/03/06
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Re: Basic Theory: -Expected Value- [Re: bozlax]
#7854758 - 10/30/06 11:09 AM

good post. expected value seems to confuse a lot of people on this forum.

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2mb
member

Reged: 09/03/06
Posts: 111
Loc: Netherlands
Re: Basic Theory: -Expected Value- [Re: bozlax]
#7854788 - 10/30/06 11:13 AM

Quote:

Board: 9c 3d 6h

equity (%) win (%) tie (%)
Hand 1: 51.9765 % 51.73% 00.25% { 2d2s }
Hand 2: 48.0235 % 47.78% 00.25% { JJ+, AhKh, AJs, J5s, J2s, Ts7s, 52s, AJo, J5o, J2o, 52o }
(T7ss is included in this range to represent a bluff)

I guess this should be:
Quote:

Board: Jc 2h 5h

equity (%) win (%) tie (%)
Hand 1: 51.9765 % 51.73% 00.25% { 2d2s }
Hand 2: 48.0235 % 47.78% 00.25% { 55, JJ+, AhKh, AJs, J5s, J2s, Ts7s, 52s, AJo, J5o, J2o, 52o }
(T7ss is included in this range to represent a bluff)

Great post btw

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matrix
there is NO spoon

Reged: 09/04/05
Posts: 7050
Loc: UK
Re: Basic Theory: -Expected Value- [Re: bozlax]
#7854820 - 10/30/06 11:16 AM

Quote:

Quote:

Total net win of 168BB/hand.

This play is +EV and has an EV of 168BB *every* time you make it.

Two out of three of these statements are wrong. Do You See Why? Hint #1: this play is +EV. Hint #2:

Quote:

Villain(100BB) calls from the BB and both see a HU flop...

it'd be so much better if you just point out what statements are wrong and then I can correct them/you as appropriate

in my simple example numbers aren't exact, and re-reading it now I could/should have written it better to stop people nit picking at semantics...

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patchdiaz
member

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Posts: 170
Loc: Chile
Re: Basic Theory: -Expected Value- [Re: matrix]
#7854839 - 10/30/06 11:18 AM

Quote:

Flop: J 2 5 (\$1, 4 players)
Hero bets \$1, BB raises to \$3, UTG folds, Button calls, Hero raises to \$8, BB raises all-in \$24.9,Button folds, Hero calls.

[...]

Lets put his range and my hand into pokerstove and see...

Board: 9c 3d 6h

equity (%) win (%) tie (%)
Hand 1: 51.9765 % 51.73% 00.25% { 2d2s }
Hand 2: 48.0235 % 47.78% 00.25% { JJ+, AhKh, AJs, J5s, J2s, Ts7s, 52s, AJo, J5o, J2o, 52o }
(T7ss is included in this range to represent a bluff)

Shouldn't this board supposed to be Jc 2h 5h?

Nice post anyway, thanks.

Best wishes.

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4_2_it
Donktastic

Reged: 07/12/05
Posts: 18437
Loc: Trying to be the shepherd
Re: Basic Theory: -Expected Value- [Re: patchdiaz]
#7854904 - 10/30/06 11:25 AM

tl;dr Very nice post.

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bozlax
Just a big [censored] softie at heart

Reged: 12/29/04
Posts: 8848
Loc: Wookie is right
Re: Basic Theory: -Expected Value- [Re: matrix]
#7855360 - 10/30/06 12:08 PM

Quote:

...nit picking at semantics...

What semantics? I'm not even concerned about the fact that each of you puts in 4BB preflop, so when you make your play on the flop you're not looking at 100BB but 96BB (that WOULD be nit-picky).

In your hypothetical, you know that Vill...

Oh, wait, I see now. You're talking about the equity race once all the money is in, which is the same as EV since all the money is in. But you're using that to support a discussion about making +EV "plays", while in your hypothetical you've already made your play. So, it would be better to say that your read of Vill is that he'd only make this sort of play with KK/QQ/JJ/TT (with AA or a set he'd play the flop differently) so you "know" you're a 92:8 favorite (actually 91:9, since his range has more cards that'll make his set) making the EV of a call:

EV = (.91)(96) + (.09)(-96) = 78.72

Right?

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ama0330
more whining, less poker

Reged: 04/20/06
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Re: Basic Theory: -Expected Value- [Re: bozlax]
#7855557 - 10/30/06 12:26 PM

*Bookmarked*

nh

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RunyonAve
saving BBV one lesbian at a time

Reged: 02/18/05
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Re: Basic Theory: -Expected Value- [Re: ama0330]
#7856074 - 10/30/06 01:07 PM

Great Post.

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kaz2107
Jumpman HOLLA!

Reged: 07/26/05
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Re: Basic Theory: -Expected Value- [Re: matrix]
#7856196 - 10/30/06 01:16 PM

Quote:

Quote:

Quote:

Total net win of 168BB/hand.

This play is +EV and has an EV of 168BB *every* time you make it.

Two out of three of these statements are wrong. Do You See Why? Hint #1: this play is +EV. Hint #2:

Quote:

Villain(100BB) calls from the BB and both see a HU flop...

it'd be so much better if you just point out what statements are wrong and then I can correct them/you as appropriate

in my simple example numbers aren't exact, and re-reading it now I could/should have written it better to stop people nit picking at semantics...

yea stop tha hatin... np

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_PokerStudent_
stranger

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Re: Basic Theory: -Expected Value- [Re: RunyonAve]
#7856268 - 10/30/06 01:23 PM

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kidpokeher
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Re: Basic Theory: -Expected Value- [Re: ama0330]
#7856718 - 10/30/06 01:57 PM

Quote:

*Bookmarked*

nh

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matrix
there is NO spoon

Reged: 09/04/05
Posts: 7050
Loc: UK
Re: Basic Theory: -Expected Value- [Re: bozlax]
#7857563 - 10/30/06 03:02 PM

Quote:

In your hypothetical, you know that Vill...

Oh, wait, I see now. You're talking about the equity race once all the money is in, which is the same as EV since all the money is in.

Bingo - I have edited my OP (via the help of 42it - thanks ) the idea is simply demonstrate EV simply in a known hand example before throwing up an example where we use a range of hands and things get a little more complicated.

thanks for pointing out the errors - and the 2 people who spotted that in example #2 my board was mistyped in pokerstove which kind of skewed the results (but only by 58BB or so )

I have proofread several times now and think I got it right this time.

I hope this makes things a little clearer now.

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ajmargarine
old school

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Re: Basic Theory: -Expected Value- [Re: matrix]
#7857671 - 10/30/06 03:12 PM

nice post sir.

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bozlax
Just a big [censored] softie at heart

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Re: Basic Theory: -Expected Value- [Re: matrix]
#7858252 - 10/30/06 04:00 PM

Quote:

I hope this makes things a little clearer now.

My concern still exists: if you're talking about the EV of calling his all-in, even if you specifically know his hand, you don't include the money you're putting into the pot in your calculations. If you're talking about the equity race AFTER you've made the call, you include the full pot.

To put it another way (and this is why I included a hint in my first reply) neither your net win nor your EV for a given hand can be bigger than the amount of money put into the pot by players other than yourself.

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EMc
The People's Mod

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Re: Basic Theory: -Expected Value- [Re: ama0330]
#7858290 - 10/30/06 04:02 PM

Quote:

*Bookmarked*

nh

Stickied

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matrix
there is NO spoon

Reged: 09/04/05
Posts: 7050
Loc: UK
Re: Basic Theory: -Expected Value- [Re: bozlax]
#7859684 - 10/30/06 05:39 PM

Quote:

Quote:

I hope this makes things a little clearer now.

My concern still exists: if you're talking about the EV of calling his all-in, even if you specifically know his hand, you don't include the money you're putting into the pot in your calculations. If you're talking about the equity race AFTER you've made the call, you include the full pot.

To put it another way (and this is why I included a hint in my first reply) neither your net win nor your EV for a given hand can be bigger than the amount of money put into the pot by players other than yourself.

So to be perfectly correct our EV for calling in the AA hand example is 68BB not 168BB - but otherwise OK?

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bozlax
Just a big [censored] softie at heart

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Re: Basic Theory: -Expected Value- [Re: matrix]
#7860758 - 10/30/06 06:54 PM

Quote:

So to be perfectly correct our EV for calling in the AA hand example is 68BB not 168BB - but otherwise OK?

Aside from that minor difference, yeah.

(Actually, the EV of that call is 80.64.)

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lexxor

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Re: Basic Theory: -Expected Value- [Re: matrix]
#9997757 - 04/17/07 06:51 PM

Excellent, a new basic i put into my brain. *bookmarked*

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fees
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Re: Basic Theory: -Expected Value- [Re: lexxor]
#9998220 - 04/17/07 07:17 PM

SKLANSKY BUX LDO

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Riverdale27
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Re: Basic Theory: -Expected Value- [Re: fees]
#11536675 - 08/05/07 06:16 AM

I have not yet seen the right answer for the first example: AA vs KK. I see a lot of calculations, but they are all wrong.

You have to call an all in on the flop. Now the only thing that matters when talking about EV, is the expected value of the call. The question is correctly asked: "What is the EV of calling knowing we are against specifically KK with our hand, AA?"

And here is where the mistake is made.

Preflop this is the action:
- SB posts 0.5 BB
- BB posts 1 BB
- CO raises 4 BB
- SB folds
- BB calls 3 BB

So there are 8.5 BB in the pot when the flop comes.

The flop comes 936

And now our opponent will move all in with the kings, and tells us what he have, so we are sure he has KK

We can call or fold. Ofcourse we will call... but what is the EV of this call?

Well...

There were 8.5 BB in the pot already, and now the villain put in his remaining 96 BB. This results in a pot of 8.5 BB + 96 BB = 104.5 BB.

Now here is the situation for us: we can call and win, or we can call and lose. Winning will happen 91.6162 % of all times. Losing will occur 8.3838 % of all times.

And now pay attention very closely: When we win, our profit is not 100.5 BB (0.5 BB from the SB plus 100 BB from the villain)! When we win, we will win 104.5 BB. How is that? Well ofcourse we will win the 100 BB from the villain, and the 0.5 BB from the SB... but we will also win the 4 BB that we put in ourselves preflop.

And this is a tricky thing to understand... this 4 BB is ours, so why do we win it? Well, you decided to raise 4 BB, and from that moment on, the 4 BB is not yours anymore, it is in the pot, and you can't take it back. This is a concept that in economics they call "sunk costs" meaning costs that result from decisions from the past, costs that can not be changed anymore. And the 4 BB that you raised are exactly that. They are not yours anymore, so they become a part of your potential profit.

If we lose on the other hand, we will actually be 100 BB down in that hand... but we are talking about the EV of the call here... and the call is 96 BB. So we lose 96 BB on the call, and not 100 BB.

These concepts are VERY important to understand!

So, know that we know that, what is our EV? Well:

EV = (0.916162)(104.5 BB) + (0.083838)(-96 BB)
EV = 95.738929 BB - 8.3048448 BB
EV = 87.4340842 BB
EV = 87.43 BB

And this is the only correct EV from the call. All the other ones you will read in this topic are wrong. On average, you will win 87.43 BB per call you make on the flop in that situation. A very profitable situation as you see, but not as profitable as the author of this topic wrongly suggests: 168 BB.

Edited by Riverdale27 (08/05/07 06:23 AM)

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Anonymous
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Post deleted by Ryan Beal [Re: Riverdale27]
#11536686 - 08/05/07 06:17 AM

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South Pole
journeyman

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Re: Basic Theory: -Expected Value- [Re: matrix]
#11537520 - 08/05/07 09:52 AM

Quote:

The actual results don't matter, as long as my range is accurate

Your decision to call the bet is the whole issue for me and that's based on the range of hands you give below isn't it?

Here's your selected range JJ+, 55, AhKh, AJs, J5s, J2s, Ts7s, 52s, AJo, J5o, J2o, 52o

But IMO some of this range (J5o, J5s, J2s, QQ+) is not as likely to be raising all-in as JJ+ for instance.

Do you agree and if so doesn't this change the EV result significantly? More to the point how do you deal with this in the heat of battle when deciding to call?

Quote:

I play using the general rule that I should never fold a flopped set for ~100BB. The reason being that no matter the flop if we can get all the money in on the flop we are almost always a favourite to win the hand at the showdown vs our opponents range of hands.

Continuing the theme of my post why do you say "we are almost always favourite to win the hand vs. our opponents range of hands"? What range of hands do you consider when making such a decision or do you automatically call all-in bets given the same hand and board?

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Fiksdal
banned

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Re: Basic Theory: -Expected Value- [Re: ]
#11537582 - 08/05/07 10:04 AM

Quote:

The Future is Here!!!

Online Spades for REAL money! Use visa/mastercard or even Paypal! Accepts USA players.

Go to your browser, and type in:

can someone ban this guy already

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Danastasio1
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Re: Basic Theory: -Expected Value- [Re: South Pole]
#12236753 - 09/25/07 06:48 PM

Quote:

Your decision to call the bet is the whole issue for me and that's based on the range of hands you give below isn't it?

Here's your selected range JJ+, 55, AhKh, AJs, J5s, J2s, Ts7s, 52s, AJo, J5o, J2o, 52o

But IMO some of this range (J5o, J5s, J2s, QQ+) is not as likely to be raising all-in as JJ+ for instance.

Do you agree and if so doesn't this change the EV result significantly? More to the point how do you deal with this in the heat of battle when deciding to call?

I was somewhat able to grasp the Galfond Dollars concept, but if I was quizzed on it I would probably fail. Does this quote above pertain to a G-Bucks situation, or not?

Edited by Danastasio1 (09/25/07 06:50 PM)

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Ditso
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Re: Basic Theory: -Expected Value- [Re: Danastasio1]
#12817230 - 11/05/07 11:36 AM

great post
thanks for clearing things out

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DickieBets
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Re: Basic Theory: -Expected Value- [Re: 4_2_it]
#12821510 - 11/05/07 04:58 PM

I really enjoyed this post and plan on trying it out on my own hands.

In trying to comprehend this, however, I'm wondering in your pocket 2s example. Doesn't you calculation make the assumption that each hand in the range you listed have equal weight ?

What I mean is, I would expect pocket 5s or pocket Js to call 100% of the time, and therefore I'd lose my stack 100% of the time in that case, but I could see A-J or A-Ks rarely calling all-in or at least significantly less than 100 %. Certain players might also fold an overpair quite often, maybe even two pair. In these cases, I wouldn't necessarily win his entire stack.

While it might not matter with a set, I could imagine that when calculating EV with an overpair or possibly 2 pair it might look +EV but actually be -EV in reality?

Am I understanding correctly or am I over complicating it ?

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