SUPER SUPER BASICS: Expected Value and Pot Odds

Ok, so I was trolling through a post and ragnar4 noticed that there wasn't a topic posted and stickied about pot odds or expected value. The two really go hand in hand, but expected value is really the backbone of everything discussed in this forum so I'm going to get off my ass and actually post something that people who know less than me can benefit from as opposed to just leaching like I usually do...

First, what is +EV?

When an action is "+EV", it means that if we were to perform this action multiple times, we should have an overall net gain. The easiest example is applied to say a coin flip. If I were to offer you a deal saying that everytime I flipped a coin, if it came up heads i would give you $10 but everytime it came up tails you would give me $9, would you take it? Of course! Because you would have an overall net gain given multiple trials. How do we know this? The answer is simpler than it seems. All we have to do is multiply the probability of winning by the amount we stand to earn, and subtract the probability of losing by the amount we stand to lose and the result is our expected value. More simply, since a (fair) coin flip will land heads exactly 1/2 times, the calculation is just
EV= p(W)*gain - p(L)*loss where p(W) is the probability of Winning and p(L) is the probability of losing
EV=1/2 * 10 - 1/2 * 9
Ev=5-4.5 = .5
So for every flip we take, we expect to have a net increase of $.5 per flip!

Ok, that's great and all, but how does this apply to poker?

Well, it applies in many different scenarios but the most basic is when calculating pot odds. When someone says that they didnt offer you the correct pot odds to call, what they are saying is that the amount of money you had to call to see the next card(s) was too large compared to the chance you had of hitting your desired card. This is easier to explain using an example.

For the purposes of this example, we are going to assume that HERO has JhTh, and that VILLAIN has AcKd and both hands are face up.
The board is 2h8hKc3s with a pot of $.40
Villain bets out $.20, should we call him?

Since both hands are face up, we will get no future money out of villain when the river comes, as he will simply fold to any bet we make. So the question is, are we getting the proper pot odds to call this bet? I.E., is calling this bet +EV???

well, we go back to our simple equation
EV=p(W)*gain - p(L)*loss
EV=9/45 * $.60 - 36/45* $.20 (it is out of 45 since we know villains hand)
EV=$.12 - $.16 = --$.04

So the pot is laying insufficient odds to justify the call, and it is thus -EV. Put another way, if we called this bet over a large sample size, we would expect to lose money!

I'll show a more typical example now and explain how we can justify the 2 and 4 rule.

Lets say we now have 7h6h on a board of 5h4hKcQd. We are certain that if we hit either a heart, or a 3 or a 8 that we will have the winning hand. The current pot is $1, and villain bets $.5. Do we call?

What do you think we do? Well, the long hard way is doing this EV calculation.
EV=p(W)*gain - p(L)*loss
EV=15/47*$1.5 - 32/47 *$.5
EV= $.48 - $.34 = $.14

So we should clearly call. But this is annoying to do at the table, and we are really only interested if a call nets a positive or negative expectation. So, an easier way is consider the problem involves what the pot needs to be for an EV of 0.
EV=p(W)*gain - p(L)*loss=0
p(W)*gain = p(L)*loss
p(W) = p(L)*loss/gain

p(W) = 15/47 = 31%, which is almost twice the number of outs!
Why is that so? Well, it is because 47 is so close to 50. 15/50 would be exactly 30, so it makes sense that shrinking the denominator would make our percentage a little bit higher than twice the number of outs. This is the back bone of the 2 and 4 rule. Say we wanted to know the probability of hitting a straight draw with 2 cards left to come though, then what is the probability? Well, in this case it is easier to determine the probability we dont win. 1=p(W) +p(L) so p(W)=1-p(L). In this case, p(L) = 40/48 * 39/47 =.69
Then p(W) = 1- .69 = .31. PERFECT. we had 8 outs, and 4*8 = 32%! So with two streets left, the probability of hitting is about equal to the number of outs multiplied by 4! Thus the rule of 2 and 4 explains that our chance of winning, is just the number of outs multiplied by 2 for 1 card remaining, and by 4 for 2 cards remaining.

Lets go back to our EV=0 equation then.
p(W) = p(L)*loss/gain
We already know that p(L)= 1-p(W) so we can instead say
p(W) = (1-p(W)) (loss/gain)
p(W)=loss/gain -p(W)loss/gain
Then p(W) + p(W) L/G = L/G
p(W)(1+L/G) = L/G
p(W) =L/G/ (1+L/G)
Let 1=G/G
Then p(W) = (L/G) / (G/G + L/G) = (L/G) / ((G+L)/G)
This is the same as L/(G+L)
Then, for a call to net 0, we have p(W) = L/(G+L)

Say we had our open ended straight flush draw on the turn, and villain bets $2 into a pot of $6. To be +EV, we need our p(W) to be greater than L/(G+L). So what do we do?
Well by the rule of 2 and 4, 15*2=30%
so 30% > 2/(6+2) = 25%
So 30%>25% so this call is plus EV. This calculation using the 2 and 4 rule is much simpler and more useful than our EV calculation, and can even be done at the table!

Well, this is really my first really contribution to the forum, so i hope this is somewhat helpful to the newest of players.

There is a pot odds page, but more info explained in different ways is always better.
http://www.flopturnriver.com/start_pot_odds.html
Thanks for posting!

Originally Posted by JKDS

For the purposes of this example, we are going to assume that HERO has JhTh, and that VILLAIN has AcKd and both hands are face up.
The board is 2h8hKcQs with a pot of $.40
Villain bets out $.20, should we call him?

Since both hands are face up, we will get no future money out of villain when the river comes, as he will simply fold to any bet we make. So the question is, are we getting the proper pot odds to call this bet? I.E., is calling this bet +EV???

well, we go back to our simple equation
EV=p(W)*gain - p(L)*loss
EV=9/45 * $.60 - 36/45* $.20 (it is out of 45 since we know villains hand)
EV=$.12 - $.16 = --$.04

So the pot is laying insufficient odds to justify the call, and it is thus -EV. Put another way, if we called this bet over a large sample size, we would expect to lose money!

I is drunk but it looks like you forgot 5 outs (9c,9s,9d,As,Ad)

Is there a much easier calculation to do. This is alot to do if your playing a game. It just seems overwhelming all the math.

Originally Posted by OhBollocks

I is drunk but it looks like you forgot 5 outs (9c,9s,9d,As,Ad)

crap. i changed the example last minute lol. fixed

Originally Posted by mrhappy333

There is a pot odds page, but more info explained in different ways is always better.
http://www.flopturnriver.com/start_pot_odds.html
Thanks for posting!

aw well, still good i guess.

as for having an easier calculation, at the end it simplifies down to just knowing the 2 and 4 rule and knowing how much the opponent bet and how big the pot is. Now, its not always simple division, but if we had to calculate 31/40 we dont have to know the exact answer, knowing that this is close to 75% is usually enough.

I might add an implied odds, fold equity, and bluff explanation later too...so long as i cant find it lol

Gogo contributaments

Originally Posted by mrhappy333

Is there a much easier calculation to do. This is alot to do if your playing a game. It just seems overwhelming all the math.

2 and 4 rule. Multiply your outs by 2 on each street for a percentage or multiply your outs by 4 on the flop to get your % by the river.

Close enough. Just remember to only count outs to win the hand. Counting 8 outs to a straight when opponents are drawing to a flush can be hazardous to your stack.

Originally Posted by Example

$0.02/$0.05 No Limit Holdem
6 players
Converted at weaktight.com

Stacks:
UTG Hero ($6.31)
UTG+1 ($5.71)
CO ($10.91)
BTN ($3.19)
SB ($5.18)
BB ($6.00)

Pre-flop: ($0.07, 6 players) Hero is UTG
Hero raises to $0.25, 2 folds, someone calls $0.25, 2 folds

Flop: ($0.57, 2 players)
Hero checks, someone bets $0.35, Hero ?

ok, for example here we have a gutshot.
4 jacks * 4 = 16% to hit by the river.
theres approx .95 cent in the pot
its .35/95= aprox 2.8
so were getting almost 3 to 1 to call correct?
so now what?

Originally Posted by mrhappy333

Originally Posted by Example

$0.02/$0.05 No Limit Holdem
6 players
Converted at weaktight.com

Stacks:
UTG Hero ($6.31)
UTG+1 ($5.71)
CO ($10.91)
BTN ($3.19)
SB ($5.18)
BB ($6.00)

Pre-flop: ($0.07, 6 players) Hero is UTG
Hero raises to $0.25, 2 folds, someone calls $0.25, 2 folds

Flop: ($0.57, 2 players)
Hero checks, someone bets $0.35, Hero ?

ok, for example here we have a gutshot.
4 jacks * 4 = 16% to hit by the river.
theres approx .95 cent in the pot
its .35/.95= aprox .38
so were getting almost 3 to 1 to call correct?
so now what?

edited it cuz its $.95 not $95.

So 16% < 38% so we are getting insufficient odds to call! What this means is this: should we call this bet many times over a large sample, we expect to lose money. IE, the times we do hit does not win us enough money to counter all the times we missed and lost the bet we called.

Mind you, in this scenario we are out of position and are unlikely to see a free river. Therefore, we are probably going to have to call a bet on the turn as well. Because of this, we should use the rule of 2 instead, meaning we have only 8% to hit the turn. Also, we are assuming we're beat and that there is no implied value.

Heh this is the kind of stuff I was posting about and doing calculations on when I first came to this site. Pot odds paint kind of a grim picture though, unless you have a monster draw they will imply you can't call a heck of a lot since people aren't routinely betting under half pot. Implied odds is what makes everything work.

Recently I've been wondering though how useful it really is to know these kinds of calculations. Incidental evidence is that a friend of mine has been living off of poker for the last 5 years, and he can't calculate even simple pot odds. He does know some simple odds now (from me) but that's it. The human mind mainly works with pattern recognition, so you'll implicitly learn these odds through trial and error, without knowing what they really are. (my thesis now is in the field of artificial intelligence, so this has some interest to me - there was actually a thesis on writing a poker bot, but sadly I was too late to get that one)

Anyway not very relevant, nice post you made, just some side thoughts on the subject of EV calculations etc.

Ok..

So lets expand our mind a little bit. Pot odds really shine when talking about limit, but man they are a sad sorry state of affairs when we look at them here at no limit.

But there's a few really awesome things we can do to increase our pot odds.

Is there any way we can increase our happyness factor when looking at this? Wouldn't it be awesome if our opponent thought. "Aww heck, I'll let this guy see the river for free." Is there anything we can do besides buying him a beer to encourage him to do this for us?

Also, lets say you did hit your hand. This isn't a videogame; once you've achieved your objective the level isn't over. It's bonus time. How can we go about earning that bonus?

What if your opponent is like my Dad, and won't call a really big bet.. even if he has the nuts. True story, he folded the nut straight on a grossly rainbow and pretty uncoordinated flop like 965r while he held 87. Guy bet 3x the pot and he hit fold before I could choke out the words "aack you have the nuts". Is there anything we could do to help encourage this type of boneheaded play?

--note. This last example is not to say I'm trying to induce misclicks... at all. Very few people are going to make such a huge mistake. Turns out though, people are prone to making small mistakes quite often. How can we induce a small mistake like the example above?

also good is http://en.wikipedia.org/wiki/Pot_odds

Originally Posted by JKDS

Originally Posted by mrhappy333

Originally Posted by Example

$0.02/$0.05 No Limit Holdem
6 players
Stacks:
UTG Hero ($6.31)
UTG+1 ($5.71)
CO ($10.91)
BTN ($3.19)
SB ($5.18)
BB ($6.00)

Pre-flop: ($0.07, 6 players) Hero is UTG
Hero raises to $0.25, 2 folds, someone calls $0.25, 2 folds

Flop: ($0.57, 2 players)
Hero checks, someone bets $0.35, Hero ?

ok, for example here we have a gutshot.
4 jacks * 4 = 16% to hit by the river.
theres approx .95 cent in the pot
its .35/.95= aprox .38
so were getting almost 3 to 1 to call correct?
so now what?

edited it cuz its $.95 not $95.

So 16% < 38% so we are getting insufficient odds to call! What this means is this: should we call this bet many times over a large sample, we expect to lose money. IE, the times we do hit does not win us enough money to counter all the times we missed and lost the bet we called.

Mind you, in this scenario we are out of position and are unlikely to see a free river. Therefore, we are probably going to have to call a bet on the turn as well. Because of this, we should use the rule of 2 instead, meaning we have only 8% to hit the turn. Also, we are assuming we're beat and that there is no implied value.

I'm mainly a lurker on this forum but am going to try to start participating from time to time. I'm a beginner at no limit though I've played for years at limit.

I have some issues/questions about the above logic:

1) We make our decisions street by street in poker. If we're just making a decision on whether to call or not, we should just look at the odds of hitting our card on the next street. So we should be using the rule of 2 (not 4).

2) Minor point: the pot is 92 cents.

3) You can't determine if a move is EV with the procedure you've outlined. I think you are mixing up odds with probabilities. In your example you get the right answer however but mainly because it is so far off from being an EV call.

Someone asked if there was an easier way. One way is to just memorize a table of necessary pot odds versus outs. Another quick and easy way is:

1) Multiply your outs by 2. For example if your outs are a gutshot, you have 4 outs....so we have the number 8. Our probability of hitting the gutshot on the next street is approximately .08 = 8/100.

2) This is the only hard part. We want to represent 8/100 as a number like 1/x. To do this we need to divide 8 into 100. Can you do this in your head? The answer is 1/12.5.

3) The pot odds that you need is then x-1....or in our example here, 12.5-1=11.5

So, to call and try to hit a gutshot you need around 11.5 to 1 pot odds. In the original example, you needed to pay 37 cents to win a pot of 92 or the pot odds were 2.5 to 1.....not nearly enough.[img][/img][img][/img]

Originally Posted by borges

1) We make our decisions street by street in poker. If we're just making a decision on whether to call or not, we should just look at the odds of hitting our card on the next street. So we should be using the rule of 2 (not 4).

2) Minor point: the pot is 92 cents.

3) You can't determine if a move is EV with the procedure you've outlined. I think you are mixing up odds with probabilities. In your example you get the right answer however but mainly because it is so far off from being an EV call.

1 was explained in the example, and a few cents one way or another is usually not going to make a difference. In fact, we can often make reduce the pot and bet a little in order to make the mental math easier without drastically changing the result. For instance, here the pot was .57 and the bet was .35. We can simplify this to .6 and .4 making our L/(G+L) = .4/1 = 40% which is about 38%. If we change these probabilities into odds, im sure you'll find that the result is close to 2.5:1.

Also, for number 3 you can't refute a bunch of math by saying "no you can't". If you are going to make a blanket statement like "you cant determine if the call is EV" you need to explain yourself, because i clearly just showed you could Additionally, as i hinted at above, during this calculation we get a comparison of percentages, ie, 8% < 40%. We can easily change these numbers into actual odds, but there is little point in doing so, as the inequality will still hold true and we still find that the odds are insufficient to justify a call.

Originally Posted by JKDS

Originally Posted by borges

1) We make our decisions street by street in poker. If we're just making a decision on whether to call or not, we should just look at the odds of hitting our card on the next street. So we should be using the rule of 2 (not 4).

2) Minor point: the pot is 92 cents.

3) You can't determine if a move is EV with the procedure you've outlined. I think you are mixing up odds with probabilities. In your example you get the right answer however but mainly because it is so far off from being an EV call.

1 was explained in the example, and a few cents one way or another is usually not going to make a difference. In fact, we can often make reduce the pot and bet a little in order to make the mental math easier without drastically changing the result. For instance, here the pot was .57 and the bet was .35. We can simplify this to .6 and .4 making our L/(G+L) = .4/1 = 40% which is about 38%. If we change these probabilities into odds, im sure you'll find that the result is close to 2.5:1.

Also, for number 3 you can't refute a bunch of math by saying "no you can't". If you are going to make a blanket statement like "you cant determine if the call is EV" you need to explain yourself, because i clearly just showed you could Additionally, as i hinted at above, during this calculation we get a comparison of percentages, ie, 8% < 40%. We can easily change these numbers into actual odds, but there is little point in doing so, as the inequality will still hold true and we still find that the odds are insufficient to justify a call.

Let

PW=probability of winning
P=size of pot
B=bet size you need to call

Then to determine if a call is +EV you need to check if the expected value of your action is positive. Thus

(PW)xP - (1-(PW))xB > 0 for a positive EV call: Performing some
simple algebra we can get the above equation into

PW > 1/( (P/B) + 1)

If we express PW in the form 1/x then we get my methodology
if x-1 is less than P/B, we should make the call.

I didn't realize that you were doing various other round off operations.
My profuse apologies. Upon my first reading, I thought you were comparing the pot odds directly to the probability of winning.....

PW > 1/( (P/B) + 1)

is exactly the same as p(W) > B/(B+P) DUCY?
My original final solution is equivalent to this, though your variable names are more convenient.

Then all we need to do, is simply determine p(W) by rule of 2 or 4, and know what the bet and the pot is and we can easily see of the call is +ev.

Originally Posted by JKDS

PW > 1/( (P/B) + 1)

is exactly the same as p(W) > B/(B+P) DUCY?
My original final solution is equivalent to this, though your variable names are more convenient.

Then all we need to do, is simply determine p(W) by rule of 2 or 4, and know what the bet and the pot is and we can easily see of the call is +ev.

OK........it's just that you in the example are using a different concept
of what the Pot is than is expressed in your (our) equations....DUCY????

my head is gonna asplode!