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 Post subject: Probability (1)Posted: Tue, 16 Aug 2011 02:21:08 UTC
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A friend of mine on IRC recommended the following problem:

Let be the size of the largest family of independent events in a finite probability space with n points (the uniform probability measure is assumed). What is the value of ?

For n=0 the answer is 1 trivially as all we have is . For n=1, the answer is 2, and similarly for n=2 we have the maximal family .

Edit: Correction as per outermeasure's observation.

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 Post subject: Re: Probability (1)Posted: Tue, 16 Aug 2011 02:31:41 UTC
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A friend of mine ......

You got a friend

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 Post subject: Re: Probability (1)Posted: Tue, 16 Aug 2011 06:43:30 UTC
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A friend of mine on IRC recommended the following problem:

Let be the size of the largest family of independent events in a finite probability space with n points. What is the value of ?

For n=0 the answer is 1 trivially as all we have is . For n=1, the answer is 2, and similarly for n=2 we have the maximal family .

Independent according to which probability measure? (My guess is the uniform distribution on n points?) Why isn't included in your n=2?

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 Post subject: Re: Probability (1)Posted: Tue, 16 Aug 2011 06:46:49 UTC
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outermeasure wrote:
A friend of mine on IRC recommended the following problem:

Let be the size of the largest family of independent events in a finite probability space with n points. What is the value of ?

For n=0 the answer is 1 trivially as all we have is . For n=1, the answer is 2, and similarly for n=2 we have the maximal family .

Independent according to which probability measure? (My guess is the uniform distribution on n points?) Why isn't included in your n=2?

Yes uniform is the right way to go. And you're right, {1) should be there. My friend is usually very good with probability, so I didn't really think about checking his examples. I'll fix the original post.

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 Post subject: Re: Probability (1)Posted: Tue, 16 Aug 2011 07:21:08 UTC
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Spoiler:
prime divisor counting function (with multiplicity)

Am I on the right track here? Hmm... looks like I am.
Spoiler:
Consider those nontrivial proper subsets --- there are at most of them:
Claim: For each prime p such that , , we can have at most e subsets that have size , with and f<e.
Proof: If there are more than e of them, the product of probabilities would be with b>0, s and t both coprime to p, contradicting the probability must be of the form .
Now throw in the empty set and the whole space, giving a total of independent events, for all .

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