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Consider the set of sequences of zeroes and ones of length $N$ with $k$ ones (or, Np ones where $p=k/N$). We draw randomly and uniformly a sequence from this set.
I want to show that with probability tending to $1$ as $N→∞$, there are approximately $kN/2$ (or $Np/2$) ones in the first half of this sequence.
$\begingroup$@Mark: I also posted this question in mathexchange (after Dougkas comment). Robert Israel also asked this question, and my answer: You right, in the case of irrational $p$ we can't have exactly $Np$ $1$'s. However, for very large $N$ we can approximate it with "error" small as we want. By the way, in this site : j.ee.washington.edu/~bilmes/classes/ee515a_spring_2012/… you can find a lecture notes in which there are some nice properties of type class (starting from page 13 in the pdf), which maybe will help.$\endgroup$
