Something I've been thinking about

[Ratamahatta]

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Since the distribution is the same...

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What do you mean by that? Distributions of those 5 examples are not the same...

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Hero has 50% equity against villain's stated range in all 5 examples.

When we know the villain's range with 100% certainty, the only thing that matters is our equity against that range. The composition of that range is irrelevant.

The composition of villain's range generally DOES matter, because we aren't 100% confident in that range. We need to analyze, even if only by gut feel, how sensitive our equity is to small changes in the range.

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Yea, ignore my comment, I am hallucinating. Same mean, same distribution, same variance.

[blah-blah-blah]

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Since the distribution is the same and the overall equity is the same, my first impression is that the variance will actually be the same for all of these situations. That is, a coin flip is a coin flip, no matter how complicated the "coin" is. In all cases, we have a sample size of a single hand, which we will win or lose, where we have 50% equity. There is no difference.

What am I missing?

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say our equity vs. his range is 50%. however say we both have the same hand w/ no redraws. our equity is and always is regardless of what card comes, 50%. this means no variance since we're always chopping the pot. if his hand range includes two hands one w/ 0% and one w/ 100% there will be variance because the results are different.

[LearnedfromTV]

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i think what matters as to which of the 5 examples i want will depend very heavily on villain, the action to the point in the hand when the equity is being evaluated, and the way the equity will develop over future streets, if there are any.

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At least as far as deciding how to play the hand, and how profitably you can play the hand.

Generally, I think I would prefer a situation where there are a variety of equities, most of the equities are relatively close to 50%, stack/pot ratio is high, and I will be better able to evaluate the effect of future cards on the equities than villain (which may be a function of my hand, his range, our relative skill, or a combination).

If you're simply asking this as a variance question, I don't think it should matter.

[Ringmaster]

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Lets for arguements sake say that although all these plays are 0 EV (ie, ignore pot size as well), you want to do all of them.

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What plays? Is this an all-in situation?

I'm going to assume that we're moving in here, and the villain is calling with one of the 5 ranges you listed (is this what you had in mind?).

If so, it doesn't matter which range he has - both the expectation and variance are the same in all 5 cases.

[Dan Bitel]

This is only a varience question.

My question is, are the varience in all 5 cases (and in particular, the ones I've asked) the same?

[ReptileHouse]

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This is only a varience question.

My question is, are the varience in all 5 cases (and in particular, the ones I've asked) the same?

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It's very, very close to the same, yes. As suggested above, there is a slight reduction in variance for the non-binary cases due to chopped pots. In practical terms, that's a small enough difference to be negligible, however.

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.

[Ringmaster]

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My question is, are the varience in all 5 cases (and in particular, the ones I've asked) the same?

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Yes.

[Isura]

I never took stats but intuitively I'd say that they go 1,2,3,4 from lowest to highest variance. Also, 2 and 3 have the same variance. My reasoning is that you can calculate the variance of each individual event and then sum using their likelyhood. So 1 has 0 variance. 2 and 3 are same since the case of 25% equity vs 75% equity are the same since you sway the same amount from the average (ie 75 or 0, 25 or 100).

[Isura]

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My question is, are the varience in all 5 cases (and in particular, the ones I've asked) the same?

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Yes.

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Isn't 1 just like taking insurance ala hellmuth on high stakes poker? So there is 0 variance

[Sorcerer808]

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A coin-flip is a coin-flip

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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.