help calculating EV

[_TKO_]

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)

[fees]

fold preflop

[drzen]

No.

[drzen]

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.

[qpw]

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.

[AaronBrown]

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.

[_TKO_]

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.

[drzen]

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).

[qpw]

[ 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.

[drzen]

[ 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.