help calculating EV
 

Let's say I'm on the river and have to decide whether to bet or check. When calculating the EV, do include my bet in the positive reward?

ie EV = (chance I win)(pot + my bet + his call) - (chance I lose)(my bet)

Re: help calculating EV
 

fold preflop

Re: help calculating EV
 

No.

Re: help calculating EV
 

If you toss a coin, a dollar a toss, you take heads every time, EV by the formula you give will be:
0.5(1+1)-0.5(1).

You'd make 50c for each toss. Good luck finding someone who'll pay you it.

Re: help calculating EV
 

Yes, you do.

What you don't do is multiply your bet by your expectation of losing before doing the subtraction.

Suppose you think you will win 1 time out of three, and the pot is $5.

Consider first, the case where your opponent folds:

EV = 1/3(5 + 1) - 1 = 1/3(6) - 1 = 1

Verify this by looking at a representative set of outcomes:

Instance 1: Bet $1, Win $6
Instance 2: Bet $1, Win 0
Instance 3: Bet $1, Win 0

Total outgoing = $3
Total income = $6
Value = $3

Value of each bet is total / number of bets = $3/3 = $1



Now let's assume he calls:

EV = 1/3(5 + 1 + 1) - 1 = 1/3(6) - 1 = 1.3333

Verify this by looking at a representative set of outcomes:

Instance 1: Bet $1, Win $7
Instance 2: Bet $1, Win 0
Instance 3: Bet $1, Win 0

Total outgoing = $3
Total income = $7
Value = $4

Value of each bet is total / number of bets = $4/3 = $1.3333


Obviously this is the averaged result of a long set of such bets.


In the case of a toss of a coin, you opponent can't fold so we have

EV = 1/2(0 + 1 + 1) - 1 = 0

Instance 1: Bet $1, Win $2
Instance 2: Bet $1, Win 0

Total outgoing = $2
Total income = $2
Value = 0

Which is pretty obviously correct.


Note than you can state the equation as:

EV = (probability of winning * pot) - (probability of losing * your bet)

which will give you the same result in the end but involves an extra multiplication which seems a little pointless.

Whilst checking this I discovered that several sites actually have erroneous information on this subject (unless my brain is fried today) which is why I have included all the workings out.

Re: help calculating EV
 

drzen and qpw are both correct, which is why this is confusing.

There are two common ways to compute the EV of a bet, and they give the same answer if done properly. Let p be the probability of winning, W the amount you profit if you win and L the amount you lose if you lose. Both W and L are measured relative to what you have before you bet. So W is the current pot plus any extra you expect the other player to contribute to call your bet, and L is your bet. Your EV can be written either:

p*W - (1 - p)*L, or
p*(W + L) - L

A little algebra should show you they are the same.

The first equation views this as a bet, you either win W or lose L. We'd call this payout W to L. The pot puts up W, you put up L and winner takes all.

The second equation views this as purchasing a chance to win, like a lottery ticket. You spend L in the hopes of winning W + L. We call this payout W + L for L.

Re: help calculating EV
 

Thanks. I had a feeling it was off but couldn't wrap my head around why.

Good to see your posts again, Aaron; I've been away from Theory too long, I think.

Re: help calculating EV
 

Aaron, his way looks to me like it will only work if you are looking at your outcome for winning. IOW, it's not useful for working out your overall EV from a range of possibilities.

It made my head hurt to read his explanation, but can he use his maths to work out his EV if his opponent has a range, some of which makes you a winner, some a loser? My way seems intuitively easier for that situation, which is much commoner in poker than coinflip-type situations (which I used only to show the OP that he can work out his EV without including his own bet).

Re: help calculating EV
 

[ QUOTE ]
Aaron, his way looks to me like it will only work if you are looking at your outcome for winning. IOW, it's not useful for working out your overall EV from a range of possibilities.

It made my head hurt to read his explanation, but can he use his maths to work out his EV if his opponent has a range, some of which makes you a winner, some a loser? My way seems intuitively easier for that situation, which is much commoner in poker than coinflip-type situations (which I used only to show the OP that he can work out his EV without including his own bet).

[/ QUOTE ]

Since, as Aaron has pointed out, the two equations are (algebraically) identical, it follows that what you can do with one you can do with the other.

I prefer the first way I demonstrated because:

1) It involves one less multiplication and one less subtraction (at the expense of an addition), which is a good thing for those whose head starts to hurt at the simplest of maths.

2) It's transparently obvious where the equation comes from if you just write down a table of expected outcomes as I did in the examples. Seeing exactly how an equation works is an enormous benefit in getting a real feeling for what is going on.

Re: help calculating EV
 

[ QUOTE ]
[ QUOTE ]
Aaron, his way looks to me like it will only work if you are looking at your outcome for winning. IOW, it's not useful for working out your overall EV from a range of possibilities.

It made my head hurt to read his explanation, but can he use his maths to work out his EV if his opponent has a range, some of which makes you a winner, some a loser? My way seems intuitively easier for that situation, which is much commoner in poker than coinflip-type situations (which I used only to show the OP that he can work out his EV without including his own bet).

[/ QUOTE ]

Since, as Aaron has pointed out, the two equations are (algebraically) identical, it follows that what you can do with one you can do with the other.

[/ QUOTE ]

Yes, I suppose you can, but I find your way hard practically.

[ QUOTE ]
I prefer the first way I demonstrated because:

1) It involves one less multiplication and one less subtraction (at the expense of an addition), which is a good thing for those whose head starts to hurt at the simplest of maths.

2) It's transparently obvious where the equation comes from if you just write down a table of expected outcomes as I did in the examples. Seeing exactly how an equation works is an enormous benefit in getting a real feeling for what is going on.

[/ QUOTE ]

Okay. Thanks for explaining. I find my way easier but I understand why you like yours.