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PhilGoetz comments on Open Thread, September, 2010-- part 2 - Less Wrong
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That's right - but the problem is not that "you can only decompose a conjunction that way when it's on the left side of the |, not when it's on the right." The reasoning is syntactically correct. The two cases are syntactically identical when you do the syntactic substitution I suggested above.
Or... hmm, I may be confused.
The two cases are syntactically identical because you have not explicitly explained how you meet the conditions that allow the decomposition you used, conditions which are not in fact met in the case of the coins. Your "syntactic substitution" removes the information that could be used to show you meet the conditions.
(Just to be sure, you did intend the original "proof" as a "find the error" exercise, right?)
Yes - but also to help me figure out how you ask whether 2 data sets are independent when they don't intersect.