In article <email@example.com>, <firstname.lastname@example.org> wrote:
> On Fri, 19 Apr 2013 08:10:55 -0700 (PDT), apoorv <email@example.com> > wrote: > > >I am not sure whether this set has been discussed earlier.Let the set > >X contain alll binary decimals x, between 0 and 1 , > >such that every initial portion of the decimal Contains not more than > >1/3 '1st'. For example , .00100... would be in the set. > >Then as far as I can make out,X is uncountable and so is [0,1]-X. Both > >are dense in each other. > >What would be the measure of these sets? > > I'm pretty sure it's pretty clear that X has measure zero. > > Given N, let x(N) be the number of N-bit sequences with > at most N/3 1's. Then it's not hard to show, from the > Central Limit Theorem if not by something simpler, > that x(N)/2^N -> 0 as N -> infinity. > >
Yes. Almost all reals in [0,1] have digits with limiting frequency 1/2 for digit 1. So Law of Large Numbers is enough, Central Limit Theorem is overkill.