Something I've been thinking about

[Isura]

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A coin-flip is a coin-flip, no matter how the problem is broken down. That's a large part of the beauty of equity calculations. They abstract away the details of individual hands and give us a broader understanding of what's going on.

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Ok, but does being all-in with 90% equity have higher variance than being all-in with 50% equity? I think that's the crux of the OP

[ReptileHouse]

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Ok, but does being all-in with 90% equity have higher variance than being all-in with 50% equity? I think that's the crux of the OP

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At the time the decision is made, we're never all-in with 90% equity. That's the key. When we decide what to do, we have 50% equity.

[ReptileHouse]

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Say you buy a lottery ticket $2, and you have 1/10000 to win $19 998. It's the only price. This situation is a coinflip but it has a lot more variance than a head-or-tail coinflip.

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Being laid the right odds to take a bet and equity are not the same thing. In your lottery example, you do not have 50% equity, even though the bet is break even.

[Ringmaster]

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So 1 has 0 variance.

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No.

Once we are called, it's true that the outcome is certain, but we don't know what hand he has. From our perspective, we win the pot 1/2 the time and lose the pot 1/2 the time.

Thus, we're flipping a coin for the pot, and this will be true for any range against which we have 50% equity. So the distribution of outcomes is the same in all 5 cases, and hence the variance must also be the same.

[Sorcerer808]

Ok look, mathematically, if I'm right, examples 2 and 3 have the same variance, which are the 2 situations where variance is higher while number 4 comes 2nd while 1 and 5 have no variance.

E(X) = SUM(px)
V(X) = E(X^2)-(E(X))^2

For 1 : E(X) = 0.5 V(X) = 0
For 2 : E(X) = 0.5 V(X) = 0.125
For 3 : E(X) = 0.5 V(X) = 0.125
For 4 : E(X) = 0.5 V(X) = 0.0625
For 5 : E(X) = 0.5 V(X) = 0

[ReptileHouse]

[ QUOTE ]
Ok look, mathematically, if I'm right, examples 2 and 3 have the same variance, which are the 2 situations where variance is higher while number 4 comes 2nd while 1 and 5 have no variance.

E(X) = SUM(px)
V(X) = E(X^2)-(E(X))^2

For 1 : E(X) = 0.5 V(X) = 0
For 2 : E(X) = 0.5 V(X) = 0.125
For 3 : E(X) = 0.5 V(X) = 0.125
For 4 : E(X) = 0.5 V(X) = 0.0625
For 5 : E(X) = 0.5 V(X) = 0

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That would be true if our decision point were after we knew which actual hand villain holds. Since we don't, we can't break it down like that.

That is, the variance is dependent on the amount of uncertainty at the time the decision is made. The underlying composition of that uncertainty doesn't matter.

[Ratamahatta]

[ QUOTE ]
Ok look, mathematically, if I'm right, examples 2 and 3 have the same variance, which are the 2 situations where variance is higher while number 4 comes 2nd while 1 and 5 have no variance.

E(X) = SUM(px)
V(X) = E(X^2)-(E(X))^2

For 1 : E(X) = 0.5 V(X) = 0
For 2 : E(X) = 0.5 V(X) = 0.125
For 3 : E(X) = 0.5 V(X) = 0.125
For 4 : E(X) = 0.5 V(X) = 0.0625
For 5 : E(X) = 0.5 V(X) = 0

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If you use E(X) = SUM(px), then you should be using: V(X)=SUM(x-E(X))^2*px.

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

[Sorcerer808]

[ QUOTE ]
[ QUOTE ]
Ok look, mathematically, if I'm right, examples 2 and 3 have the same variance, which are the 2 situations where variance is higher while number 4 comes 2nd while 1 and 5 have no variance.

E(X) = SUM(px)
V(X) = E(X^2)-(E(X))^2

For 1 : E(X) = 0.5 V(X) = 0
For 2 : E(X) = 0.5 V(X) = 0.125
For 3 : E(X) = 0.5 V(X) = 0.125
For 4 : E(X) = 0.5 V(X) = 0.0625
For 5 : E(X) = 0.5 V(X) = 0

[/ QUOTE ]

That would be true if our decision point were after we knew which actual hand villain holds. Since we don't, we can't break it down like that.

That is, the variance is dependent on the amount of uncertainty at the time the decision is made. The underlying composition of that uncertainty doesn't matter.

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I honestly didn't understand what you said but while writing my answer I found what was wrong in my reasoning. Assuming it's an all-in situation, half the time we're winning, half the time we're loosing, in every situation, it's that simple. You're right imo.

[ReptileHouse]

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half the time we're winning, half the time we're loosing, in every situation, it's that simple.

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Exactly. [img]/images/graemlins/smile.gif[/img]

[Sorcerer808]

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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?