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Cthulhoo comments on Can you recognize a random generator? - Less Wrong
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I see a possible confusion in your post, in the sense that you have apparently fused two hypothesis into one: *A is random *A is fair
The two are not equivalent: A can be a random generator with p(0)!=50% or it can have p(0)=50% but output a very predictable string, e.g. 01010101010101010101...
Going back to your example if you hypothesis is A=(random et fair) then seeing the string 0000000000 can lower your belief in the statement (not for the "random" part, but for the "fair" part"). If your hypothesis was A=(random et p(0)= 0.00000000000000001%) then the string could raise your belief instead.
Yes, I misspoke. The question is to discern between fair and biased random generators, not between random and non-random ones. As benelliott pointed out, stateless random bit generators seem to have quite unequal probability distributions of output sequences.