Something I've been thinking about

[Ringmaster]

Guys, you're making this out to be more complicated than it is. Just think about it.

We either win the pot or lose the pot each time the villain calls. Our equity versus his range tells us the likelihood of each of these events happening, and hence the distribution of outcomes. In other words, the distribution of wins and loses depends solely on our equity.

This means that, for any 2 given situations in which our pot equity is the same, the variance will also be the same.

[Ringmaster]

Also, ReptileHouse is spot on in this thread.

[poincaraux]

I don't know what's going on here, but if it's a situation where we're shoving vs. villain, and villain has a tiny clue about our range, I like situation 3 because maybe he'll fold the 25% hands.

[Ratamahatta]

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Edit: and how the hell did you get E(X)=0.5? [img]/images/graemlins/confused.gif[/img]

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sit 2 : E(X) = SUM(px) = (2/3)*(0.75) + (1/3)(0) = 0.5

Aint this right?

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If X= eather 1 or -1 (win $1 or lose $1) then
sit 2 : E(X) = SUM(px) = (2/3)*(0.75)*1 + (1/3+(2/3)*0.25)*(-1) = 0

[FishSticks]

Can someone cliff notes the math talk?

On a different note, this obviously isn't talking about river play (since a hand can't have 25% equity on the river obv) - so I know for me psychologically if I manage to get me stack in on the flop/turn drawing literally 100% dead (whic is pretty rare so w/e for the purpose of this discussion I guess), it tilts me pretty good so I'd want whatever choice makes that the least likely to happen to me.

[bilbo-san]

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i strongly prefer less variance. you guys should too, because it means you can move up faster. which is highly +EV in a meta-sense. not to mention everyone tilts, even those of you who think you don't, and reducing variance reduces tilt.

if you're getting at something deeper than that, you lost us all.

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This is half true.

Less variance also equals less POSITIVE variance.

Therefore amongst a pool of good players who play a high-variance style, a very significant portion of them will move up faster than anyone in a pool of good players who play a low-variance style.

[bilbo-san]

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There is no lottery-style variance with 100% equity and 0% equity.

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This is not true. 50% of the time we'll have 100% equity, and 50% of the time we'll have 0% equity. This is the same as having 50% equity 100% of the time.

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These two things have the same mean, but not the same variance.

[bilbo-san]

The flop and turn DO matter.

Having 50% equity vs. his hand is not the same thing as when villain has one of 2 hands, 0% or 100%.

This is because we will NOT win 50% of the time vs. the hand that we have 50% equity against.

Unless we are talking about a result set where n is infinite.

To dumb it down a bit:

When's the last time you flipped a coin 100 times and got heads 50 times?

[Isura]

Ok my head hurts. Who is right?

[Ringmaster]

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Having 50% equity vs. his hand is not the same thing as when villain has one of 2 hands, 0% or 100%.

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Ignoring any chance of a tie, they ARE the same in terms of variance. In both cases we expect to win 50% and lose 50%. The probability distribution is exactly the same.

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This is because we will NOT win 50% of the time vs. the hand that we have 50% equity against.

Unless we are talking about a result set where n is infinite.

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Well, of course a finite sample is not likely to match the underlying distribution exactly. So what? In the case where he holds 2 hands, either 0% or 100%, we would not expect to see each hand exactly half the time either in a finite sample, even though both are equally likely.

Think of it this way:

In the case where we know his range contains multiple hands whose weighted equities equal 50%, the coin is flipped before seeing his hand. In the case where we know his range has only one hand with 50% equity, the coin is flipped after seeing it.

The probability distribution is the same in either case, the only difference is when the coin is flipped (which of course depends on the information we have at hand).