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.

[BigBiceps]

Think of it this way:

if n = 1000 the probability of an absoulte difference of 1000 is near 0.

if n = 1,000,000 the probability of an absolute difference of 1000 is much > 0.

[danzasmack]

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

[danzasmack]

makes sense

[AaronBrown]

I'm not sure if everything has been resolved or not.

The long run average winning percentage converges to the expected win rate.

The long run number of wins "diverges" in the sense that the expected absolute difference between the current number of wins and future number of wins increases, the farther you look into the future.

However, the long-run number of wins can also be said to neither converge nor diverge. It's not a question of mathematical definitions, it's a question of English. If you flip a fair coin and win $1 for heads and pay $1 for tails, your expected wealth after any number of future flips is equal to your current wealth. It does not tend to zero or anything else.

[Jehaim]

I get it now. [img]/images/graemlins/laugh.gif[/img]