More precisely, if we know the proportion of 1's in a uniformly randomly chosen binary sequence of length $2N$ is $p$, then given an $\epsilon>0$ , what is the probability that the proportion of 1's in the subsequence consisting of the first $N$ digits is $p-\epsilon<p'<p+\epsilon$?
Answers with approximations and bounds are also welcome.
I suspect this is related to the law of large numbers, where we can think of 1's and 0's as denoting the trials that succeeded and failed respectively but I can't think of a precise answer. I also can't use python to check relatively small cases because it quickly becomes computationally unfeasible.