LLN Question (FGators Question)

[danzasmack]

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yeah but the nature of the stat is the opposite - it should converge over time.

also the 1.6 mil hand Ev graph should be in BBs IMO

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This is a common fallacy. It in fact diverges does not converge. I am surprised that the majority of the people at 2+2 here do not realize this.

"Misconception 2: If run large number of coin tosses, the number of heads and number of tails become more and more equal. This is incorrect, as the LLN only guarantees that the sample proportion of heads will converge to the true population proportion (the p parameter that we selected). In fact, the difference |Heads - Tails| diverges! "

http://wiki.stat.ucla.edu/socr/index...fLargeNumbers2

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This makes no sense to me.

I can see how the difference does not converge.

But how it DIVERGES i don't see.

Also - in the long run, your expected all in winning should converge to your actual, no?

[CmnDwnWrkn]

Well with actual coinflips, I think the divergence is due to the small difference is weight distribution between the heads and tails sides. The heads side is just a little bit heavier, so that would cause the total number of heads and the total number of tails to diverge over a large sample size. I don't see how "Misconception 2" applies to "coinflips" in poker though.

[danzasmack]

"A "law of large numbers" is one of several theorems expressing the idea that as the number of trials of a random process increases, the percentage difference between the expected and actual values goes to zero. "

http://mathworld.wolfram.com/LawofLargeNumbers.html

And as far as what we're comparing in the fgators thread it's proportion vs. actual - but just consider that like number of heads flipped vs. # trials * expected # of heads - which does converge

[danzasmack]

I think this site is worded very poorly.

Mathematical definition of diverge = does not converge.

http://mathworld.wolfram.com/DivergenceTests.html

MWebster definition:

"to move or extend in different directions from a common point"

[Catyoul]

You're basically not really talking about the same thing. The absolute value of difference WILL diverge. That difference divided by the number of hands/flips will converge though.

edit : and lol at thinking definitions on MathWorld are worded poorly.

[danzasmack]

no, i meant the original link. i quoted mathworld for a good definition. it's a great site.

[danzasmack]

makes sense

[alThor]

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lol at thinking definitions on MathWorld are worded poorly.

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lol at thinking MathWorld is perfect. You know it isn't just Weisstein, right?

Check out the 2nd last sentence of their Gambler's ruin page. (Refers to casinos.) The way it's written, it implies that even if casinos offered favorable games, they'd still come out ahead due to their "deep pockets". I pointed this error (or at best, poor wording) out to the maintainer years ago, but it's still there.

MathWorld is a great reference, but don't worship it.

[Catyoul]

It was a misunderstanding in the first place, and I still don't think any definitions are wrong. From what I understand from the Gambler's ruin page, it didn't say that casinos would come out ahead with favorable games but equal ones :

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Even with equal odds, the longer you gamble, the greater the chance that the player starting out with the most pennies wins. Since casinos have more pennies than their individual patrons, this principle allows casinos to always come out ahead in the long run. And the common practice of playing games with odds skewed in favor of the house makes this outcome just that much quicker.

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I agree it could be worded better.

[pzhon]

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lol at thinking MathWorld is perfect. You know it isn't just Weisstein, right?

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That there are other contributors than Weisstein is the main reason it is as reliable as it is. He's not a mathematician, and has little direct contact with most of the areas covered by MathWorld.

Early in its history, he asked me to take over or take some sort of leading role, and I declined since fixing the errors looked like too much work. It has gotten a lot better.