Help me understand Expected Value in Poker

I'm reading the book Poker Math That Matters by Owen Gaines and in the section where he's explaining how to calculate expected value in Poker he gives the following example:

You: 5 of spades and 6 of spades
community cards: K of spades, T of spades, 2 of hearts
Pot= $25

Your opponent goes all in with his $24, making the pot $49. You have enough chips to cover this bet. Should you call or fold?

We have a flush draw so we have 9 outs. If we call we'll see two cards so using the 4/2 rule (multiply by 4 if we get to see the turn and the river, 2 if just one card) we multiply 9 by 4 to get 36%. We have a 36% chance of winning this showdown.

Our pot odds are x / (x+y) where x is the amount to call and y is the pot amount before we call, so in this case it would be $24 / ($24+$49) which comes out to about 33%.

Since we want to make money we should only bet if our chances of winning the showdown are higher than our pot odds. In this case we have a 36% of winning which is higher than the 33% pot odds so the correct decision in the long run would be to call.

To calculate our EV we multiply the pot amount after we call by our chances of winning, then we subtract the cost to call. So in this case it would be .36 ($73) - $24 = $2.28. The EV in this case would be $2.28.

But why are our chances of winning only 36%, why is it just 9 outs? If our opponent just has for example J of diamonds and 2 of clubs, he only has a pair of 2's which could be beaten if the turn or river is a 5 or a 6. Using a showdown calculator it tells me I have a 47% of winning a showdown against a player with a random range of cards.

So wouldn't the EV actually be .47 ($73) - $24 = $10.31? That's a big difference and I don't understand why I would need to assume I would need the best possible hand (in this case a flush) when I could possibly win with a higher pair or two pairs, or three of a kind or the very lucky 6 high straight.

Okay I'm on my phone at the moment but in the scenario you need to be calculating your equity against your villains PERCEIVED RANGE.

You have merely included your outs to a flush with no consideration for your opponents range here. It varies dramatically. For example, your J2 example, do you feel your PPP would be shoving this hand? No, pretty much never so you can remove that from the equation.

If you wait till I am on my laptop I will go through this properly as best I can over here.

ConDeck said:

Okay I'm on my phone at the moment but in the scenario you need to be calculating your equity against your villains PERCEIVED RANGE.

You have merely included your outs to a flush with no consideration for your opponents range here. It varies dramatically. For example, your J2 example, do you feel your PPP would be shoving this hand? No, pretty much never so you can remove that from the equation.

If you wait till I am on my laptop I will go through this properly as best I can over here.

Click to expand...

Thanks, I can wait.

LetoAtreides82 said:

I'm reading the book Poker Math That Matters by Owen Gaines and in the section where he's explaining how to calculate expected value in Poker he gives the following example:

You: 5 of spades and 6 of spades
community cards: K of spades, T of spades, 2 of hearts
Pot= $25

Your opponent goes all in with his $24, making the pot $49. You have enough chips to cover this bet. Should you call or fold?

We have a flush draw so we have 9 outs. If we call we'll see two cards so using the 4/2 rule (multiply by 4 if we get to see the turn and the river, 2 if just one card) we multiply 9 by 4 to get 36%. We have a 36% chance of winning this showdown.

Our pot odds are x / (x+y) where x is the amount to call and y is the pot amount before we call, so in this case it would be $24 / ($24+$49) which comes out to about 33%.

Since we want to make money we should only bet if our chances of winning the showdown are higher than our pot odds. In this case we have a 36% of winning which is higher than the 33% pot odds so the correct decision in the long run would be to call.

To calculate our EV we multiply the pot amount after we call by our chances of winning, then we subtract the cost to call. So in this case it would be .36 ($73) - $24 = $2.28. The EV in this case would be $2.28.

But why are our chances of winning only 36%, why is it just 9 outs? If our opponent just has for example J of diamonds and 2 of clubs, he only has a pair of 2's which could be beaten if the turn or river is a 5 or a 6. Using a showdown calculator it tells me I have a 47% of winning a showdown against a player with a random range of cards.

So wouldn't the EV actually be .47 ($73) - $24 = $10.31? That's a big difference and I don't understand why I would need to assume I would need the best possible hand (in this case a flush) when I could possibly win with a higher pair or two pairs, or three of a kind or the very lucky 6 high straight.

Click to expand...

Right at my laptop now...

So as I said earlier you need to work out your equity in the pot against your opponents perceived range.

Ranging your opponent is a skill in itself that I will not go into here but I will make some assumptions based on the limited provided.

Lets say that as Villain is a short stack (only has a pot sized bet behind on the flop) his opening range pre flop is going to be fairly tight.

We will assume for this example that it is around 12% of hands.

A 12% range looks as follows:

77+,ATs+,KTs+,QTs+,JTs,ATo+,KQo,QJo

So on the board you stated above of:

Ks Ts 2h

we are able to remove some cards. Now this is again player dependent but we will make some assumptions again.

Lets, for the sake of this example, presume that villain will do this with top pair or better, any flush draw and any straight draw.

That means his range is now as follows:

KK+,TT,AKs,AsQs,AsJs,KJs+,QJs,AKo,KJo+

So as far as working out our equity goes we have to calculate this against the above range (not just how often we will make a flush).

Against this range you have roughly 35% Equity. This includes all runner runner 2P, Trips etc.

So as you described above the EV calculation would be:

.35($73) - $24 = $1.55 (lower than your original estimation)


Although you were not far off with your rough estimate above do you see how important it is to range your opponent?

Now we have used a very generic "lab" range here, this would vary significantly and this is where your skill comes in.


As you can see below tightening villains range by removing straight draws and KJ to

KK+,TT,AKs,AsQs,AsJs,KQs,AKo,KQo

means that you now only have 33% equity which means that the call is only +0.09EV.

.33($73) - $24 = $0.09

This is dramatically different, do you see now why range has an effect?

Another consideration I would make is that a fold is always neutral EV. If a play is only minor +EV I will not make this an "automatic" play there are other considerations. For example is it worth risking $24 that you could get in in a much better spot later in the game to make $0.09 LONGTERM. This means you might lose the next 12 times you make this play and then win the next four and you have still profited by $0.36 but your bankroll may not be able to withstand the variance here for such little gain and you may never see a profit.

I should also add I would not be in the above situation very often however, and neither should you. 67s is not a hand I am opening unless in LP (HJ,CO or BTN for FR) and with the short stack on my left I would tighten more as we can expect a reshove often. If the pot to stack ratio is like it is in your example due to numerous callers preflop then this dramatically changes our equations as we have to consider any person left to act after us and the above only works for a heads up pot.

Lets look at an example from my hand history that may demonstrate better and in a real life scenario:



pokerstars Zoom Hand #138872544962: Hold'em No Limit ($0.01/$0.02)

2015/08/02 22:58:55 WET [2015/08/02 17:58:55 ET]

Table 'Halley' 6-max Seat #1 is the button

Seat 1: foxsurf ($0.15 in chips)

Seat 2: endrju1993 ($2.60 in chips)

Seat 3: aaferreira76 ($3.35 in chips)

Seat 4: Kanonkulspel ($6.35 in chips)

Seat 5: fishi1996 ($3.89 in chips)

Seat 6: ConDeck ($11.17 in chips)

endrju1993: posts small blind $0.01

aaferreira76: posts big blind $0.02

*** HOLE CARDS ***

Dealt to ConDeck [Ad Jd]

Kanonkulspel: folds
fishi1996: raises $0.04 to $0.06
ConDeck: calls $0.06
foxsurf: folds
endrju1993: folds
aaferreira76: calls $0.04

*** FLOP *** [7d 5d 5h]

aaferreira76: bets $3.29 and is all-in

fishi1996: folds

ConDeck: ???

Unknown Villain so no notes or stats which makes ranging a little more difficult but we can assume the following range is pretty safe:

77+,A5s,65s,A5o,65o

I have 37% equity against the above range with my 2 overs and a flush draw.

If we use the EV calculations here

.37($6.78) - $3.29 =-$0.78

We can see this is a losing call.


I hope that helped to clarify a little around the importance of ranging. You are just making the assumption that your opponent is playing any 2 cards when calculating you EV at 47% in your example, and this is not true you can always eliminate hands and how accurately you can do this the better your decisions will be the fewer the mistakes you will make. Also you are assuming you have clean outs. If your opponent has a higher flush draw you may not be good even if you hit and if he has a set the board could pair meaning that your flush is also no good, your two pair outs are also removed.

Does this make more sense now?

Thanks ConDeck, it makes a lot more sense now. It's a good thing I asked because I would have been making a serious mistake always going with the random range for my opponents.