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Ivan Pedroso said:
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alexti said:
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Bummer_Duck said:
shouldn't it approach 3/8? or 37.5% the larger the sample is? What am I missing here?
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You mean number of mages concentrated in 3 paths should approach 37.5% of total number of mages? - No it should not, what are you saying would effectively mean that the equal number of mages in each path, which is very unlikely event. On the large samples peak of probability will probably be somewhere in 45-60% range (that's very rough estimate, I will try to calculate it precisely some time later)
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Hmmmmmm, why shouldn't it approach 3/8 ?!?
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Do you talk about convergence? Like lim(percentage, N->inf)= 3/8?
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Ivan Pedroso said:
Let us assume that the distribution behind the scenes is uniform. Then the observed frequencies will approach 1/8. Of cause getting a sequence that actually results in an observed frequency of exactly 1/8 for every path will be highly unlikely, but they WILL approach 1/8 as the sample grows.
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not WILL, only likely. For example, for any sample size, you have positive probability of getting all picks in one path. For samples of large size the distribution function will be getting more or more condensed, which you may call "approach". But you can't provide N for some small value x such that percentage of picks concentrated in 3 paths will deviate from 3/8 by no more than x.
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Ivan Pedroso said:
I mean: if you use a uniform distribution to generate some values, then the distributed values will look more and more uniform. And therefor adding the frequencies of the three highest represented paths will tend towards 3/8 (from above obviously). I concede that getting the result 3/8 in a test sample will "never" happen.
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The resulting process (the percentage of picks concentrated in 3 paths) is also a random process. That means that for different samples of the same size will produce different results.
The chances of getting 3/8 are getting smaller as you increase the sample size. The root of the issue is that the more your sample size is, the more possible outcomes can happen. That makes every particular outcome less and less likely to happen.
This random process also has a certain distribution. Which will roughly look like:
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3/8 1
If we pick the range [x1,x2] that covers 99% of possible outcomes, then we could show that lim (x1, N-> inf) = lim (x2, N-> inf) = 3/8 (well, I think we can show), which intuitively seems as "approach", but unlike convergence, this "approach" has a stochastic character, like the difference between *will* and *most probably will*