# Expected Value article by FPaulsson

#### ChuckTs

##### Legend
Silver Level
i only just read it, but great stuff, man!
as with all theoretical poker, there are arguments to be made though

the situation you stated

Quote:

"A♥ J♣
And the board shows:
A♣ 10♣ 5♦ 8♣ 3♣

You're in first position, the pot is \$100, and the big bet is \$10. Do you bet?

Let's say, for the sake of argument, that your opponent can hold any two cards and will always fold if he doesn't have a club. Let's also stipulate that he'll call a bet with any club, and make it two bets if he has the K♣ or Q♣. Let's also say that in case you check, he will bet with any club and check with no clubs.

Let's do the math. Since he can hold any two cards, each of the individual clubs is as likely to be in his hand (and let's pretend that he can't have two of them - because we know him well enough to know that he would have raised the turn if he did).

Note: We do not bother adding in the times when he has no club at all, in these scenarios. Your opponent will fold if you bet, and check if you check in these cases, and you will then always win the pot uncontested. For the mathematically curious, this actually has implications on the expected value for the situation as a whole, but not for the specific purpose that we're discussing it here: Determining the correct strategy."

i know this is just a basic situation, but of course real hands never occur exactly as theory predicts and neither do these kinds of statements apply to all kinds of players.
ie, most players won't call when bet into with a 4-flush showing if they have a [2c] or even up to [5c] or higher. Nor will the average player bet with the same hand if checked to.
There are always people who play differently, and thus you can't apply exact calculations like that to real games sometimes, could you?
just wanted to point it out

this was my first time reading about expected value applied to poker and I thought it was well done, paulsson! Nice article, I enjoyed reading it. Gave me some good insight + info in expected values
-ChuckTs

#### ChuckTs

##### Legend
Silver Level
Just reread the article and wanted to know if you could post some more examples of Expected Value, F.P.?
I'm really new to Limit holdem and would like to improve my basic skills and i felt that the EV article really helped, but i need more examples to know what else is + and -EV

also i think you may have missed my original post in this and wanted to hear your take on my opinion

#### F Paulsson

##### euro love
Silver Level
Sorry, I probably did miss it. I often go into periods of time where I have very little time left over for reading the forums, and I guess you caught one of those (the day when I would otherwise have read the post would actually have been on the day I got engaged, so I hope you forgive me for being preoccupied )

I'll try to get back to this post tomorrow or so, I landed in Beijing a few hours ago and am kinda jetlagged. Right now I'm just struggling to stay awake for long enough so I don't wake up at 4a.m.

If I haven't gotten back to you this weekend, post a reply in this thread and say "Slacker! Gimme my example!" and it'll pop up in my unread posts, just in case I don't remember any of this tomorrow (I believe in drinking on airplanes).

#### F Paulsson

##### euro love
Silver Level
Some basic ideas about +EV plays:

If you have AK preflop, and someone else has AJ, you're an 75% or so favorite (I don't actually know how well these stack up against each other, and I can't be bothered to check - the point is not in the percentages either way) - raising preflop is +EV. You expect to win 75% of every bet you make, so raising is clearly good.

Here's a fairly widely misunderstood part of EV:

If you have a flushdraw on the turn (let's say that the outs are clean, but you have no other way of winning, so you have 20% equity), and there is 10 big bets in the pot already, someone bets into you:

(based on pot odds only, not implied odds)
Folding has an EV of 0.

Calling will yield you 12 bets 20% of the time, and lose you 1 bet 80% of the time => Expected value = 12*0.2 + (-1)*0.8 = 2.4 - 0.8 = \$1.6

Raising (let's say he will always call) will yield you 14 bets 20% of the time, and lose you 2 bets 80% of the time => Expected value = 14*0.2 + (-2)*0.8 = 2.8 - 1.6 = \$1.2

Raising here has positive expected value, but it has lower expectation than calling!

I could go on with more examples, but I'm not entirely sure where to begin - everything can be broken down into EV examples.

#### F Paulsson

##### euro love
Silver Level
That was quite a messy post I just made - sorry about that, I'm operating under some time pressure as I have a meeting to prepare for. I hope it made some sense, at least.

#### F Paulsson

##### euro love
Silver Level
Back at the hotel, and I just thought I'd have a look at the semi-bluff, like the one listed above. I've changed the scenario somewhat, though.

The semi-bluff is very widely used, and I suspect that it's overused. So here's the calculation (I'm doing this "blind," e.g. I don't know what these calculations will yield).

The idea behind the semi-bluff is that we bet or raise with a drawing hand because we have two ways to win: By either making the other person(s) fold, or by catching our draw. The theory behind it is that the combined chances of these two events can make it an overall +EV play. So let's look at a common scenario and determine how often our opponents need to fold to make a semi-bluff correct.

All other aspects of this will be ignored, e.g. implied odds that are improved/worsened by us raising now. For simplicity's sake, we'll assume that if we just call, we'll be able to win 2BBs on the river (we have position, call now, and then raise on the river and get called if we hit), and if we raise, we'll only win 1 more BB on the river (1 person calls the raise on the turn, and check/call the river).

Scenario: Three people to the flop and turn. It was raised preflop, one bet (from middle position) on flop which both other players call, and now the middle position player has bet again, just as you picked up a flush draw. You are on the button.

Other assumptions: The first position player is very unlikely to call two bets cold on the turn - if we raise, we're pretty sure he will fold. Actually, let's say he folds anyway. How often does the bettor (MP) have to fold for this play to be +EV?

There are now 6SB from preflop, 3SB from flop, and 1BB from turn in the pot = 5.5BB.

The different scenarios:

You call. 20% of the time, you will spike your flush (and win 2 more bets on the river), 80% of the time you will lose 1BB. This gives you an expected value of 0.2*(7.5) + (-1)*0.8 = 1.5 - 0.8 = 0.7BB.

You raise. How often must villain fold for this to be better than 0.7BB? Let's say that he will call X percent of the time. It's clear, then, that the equation is this:

(0.2*X)(7.5) + (0.8*X)(-2) + (1-X)(5.5) = 0.7

So we solve for X.

1.5X -1.6X + 5.5 - 5.5X = 0.7
-0.1X - 5.5X = 0.7 - 5.5
-5.6X = -4.8
X = 4.8 / 5.6 = 0.85

Uh.

Okay, that actually surprised me. Apparently, for the turn semi-bluff raise to be a better move than to just call, we need to trust that the bettor will fold 85% of the time or more. I feel like my math is messed up somewhere, because this is higher than I expected.

Then again, because of messed up airconditioning (read: no airconditioning) I was awake until 4am this morning and had breakfast at 7, so chances are I'm not thinking straight and that my calculations suck. If anyone wants to check my math, feel free - I'd like to know if it really is this bad to semi-bluff raise the turn.

Cheers,
Fredrik

#### ChuckTs

##### Legend
Silver Level
I could go on with more examples, but I'm not entirely sure where to begin - everything can be broken down into EV examples.
any would help i'm not looking for specific ex's but rather just any in general to better understand the concept of EV

and p.s. the first post was fine and actually really helped - would raising a flush draw on the flop be +EV? also how would you factor in still having the turn to act on with EV?

as for the second, I understood the math fine, and can't see why it came out wrong for you

Thanks for the help Freddie - i appreciate you taking your spare time to write this out for me and the others who are reading this

#### F Paulsson

##### euro love
Silver Level
ChuckTs said:
any would help i'm not looking for specific ex's but rather just any in general to better understand the concept of EV

and p.s. the first post was fine and actually really helped - would raising a flush draw on the flop be +EV? also how would you factor in still having the turn to act on with EV?
Regarding raising on the flop as a +EV move: Generally speaking, whenever you raise for value it has to be because your hand has a higher-than-average equity (sorry, Rob!) compared to the other hands that would call - and a hand can have an above-average equity despite being only a "drawing hand." The prime example would be, say,

[Kh][Qh] on a board of [Jh][10h][2c]. There are so many cards that will give you a winning hand by the river even if you're still behind (at least 8, likely 15, and maybe 21 outs) that if you have more than two opponents who will call, a raise here will give you long term winnings. Look at it this way:

If you have a flushdraw on the flop, and you have 9 clean outs to hit your flush (by "clean" is usually meant that if you hit your flush, you're guaranteed to win - you can discount your outs a bit if you suspect that someone else may hit a full house or a higher flush, but that's not the topic for today), then you're ~35% chance to hit your flush by the river. Conversely, you have 35% equity in this hand on the flop. Or, put differently again, you will win 35% of all the bets that go in on the flop. If you have two opponents that will call you, then you will bet, let's say, \$1 and they will each pay \$1 as well. \$3 went in on the flop, of which you're expected to win 0.35*1 = \$1.05. You've shown a 5 cent profit. Note that you have to have at least two opponents for this to be worth it.

There are some tactical drawbacks to raising the flop with a flushdraw, however, and although this was primarily a discussion about EV, let's have a look at it anyway:

Anyone who has read Small Stakes Hold 'em by Ed Miller is likely to have picked up that raising with strong draws on the flop is +EV - but that's strictly mathematically speaking. Often, raising on the flop will increase your profit on the flop but will severely hurt your implied odds for when you actually hit your flush. Look at this scenario:

You have QTs, and your opponents holds KJ and AJ, let's say. The flop comes

J-9-3, two of your suit. You have an open-ended straightdraw, and a flush draw. You're on the button, and the first player - with AJ - bets. The second player raises. Should you 3-bet?

Well, clearly you have a +EV decision if you 3-bet here. Your equity in this pot is well over the 33% you need (as you have around 14 outs, if none of the other hands hold your suits), but is it a good way to make money? Not necessarily.

If you raise, the other two will get suspicious of a set or two pair. If you just call, it's possible that the first player will 3-bet and take back control of the hand - what will then happen if you hit your monster on the turn?

He will bet out again, the second player will call (most likely) and then you can raise - hitting them for another BB each! If you raise the flop, you're unlikely to be bet into on the turn, so your implied odds for when you do hit your hand will likely go way down. However, it's worth noting that if you do 3-bet the flop, you can probably go for a free card on the turn if you don't hit your hand there, which is nice. But specifically when people say that you raise for value with a draw on the flop, this may not be a good situation to do it. Consider first how the later streets will play out if you raise now, and make the best decision.

If this had been a flop with several callers who like to "peel the flop" for one small bet but are likely to fold on the turn unimproved, then raising the flop is a good play, though.

as for the second, I understood the math fine, and can't see why it came out wrong for you
I don't have time to check it again now, but I'm not sure my math is wrong - I was just surprised by the result.

Thanks for the help Freddie - i appreciate you taking your spare time to write this out for me and the others who are reading this
My pleasure! Keep in mind that I have selfish reasons for writing this as well; doing the calculations teaches me stuff as I go along - and I think I learned yesterday that semibluff betting/raising the turn in a 5.5BB pot is a bad idea.

#### ChuckTs

##### Legend
Silver Level
ok; was bored at home and decided to run through some EV situations and I know i'm doin something wrong, because all the situations (bet, check-raise,
etc) have either all +EV or have the same EV...very confusing for me.

I'm going to try and write this down as i had it on my sheet, and if anyone could help me (nudge nudge FP ) as to what i'm doin wrong, that'd be great.

Limit Holdem:
In BB, it is folded around to the button who raises, its folded by SB and you call (for whatever reason you don't reraise) with TT
To give myself some kind of sense as what my equity is in this hand, I give the Button a set of hands that he would raise with:
AK through A4, KQ, KJ, KT, QJ, QT, JT 9T and all the pairs down to 77 (so 25 hands)

SO... the flop comes 8-3-2 with 4.5 SB in the pot
I have 72% equity here (only 7/25 possible hands that button is playing can beat me at this point)
to make it very straightforward, i've set the rule that the button will raise if he's ahead, and fold if he's not (if he raises me, i will fold)

so here are two options:

bet: (0.72*5.5 SB) - (0.28*1) = +3.68

check-raise: (0.72*7.5 SB) - (0.28*2 SB) = +4.84

is this correct?
i'm following this formula (can't remember if it's correct or not)
(equity * [your bets that round + pot]) - ([%100-equity] * [your bets that round])

#### F Paulsson

##### euro love
Silver Level
Without checking your calculations, I can say right off the bat that the sum of the expected values for all your actions is not a zero-sum (or doesn't have to be). When you have a better than 50% equity in a pot, no action will be -EV. Folding is has an EV of zero, and betting and raising will always be +EV. You should always strive to find the action that has the HIGHEST expected value, though, and that can involve choosing between several +EV options, or choosing the lesser of several -EV decisions.

Now for your calculations: They look correct to me.

The formula you're using fits what you're trying to do, yes. The formula for expectation is more general than that, but for this purpose, yours is close enough.

#### ChuckTs

##### Legend
Silver Level
oh yeah another thing is i calc'd EV for having a naked Ax flush draw with a big card on board and came up with all +EV options