It is easy to construct $A$ for which the limit does not exist. Consider the following set $A$. Include all even numbers in the intervals $[10^n, 10^{n+1}]$ for even $n=0,2,...$, and all numbers divisible by 3 in the intervals $(10^n,10^{n+1})$ for odd $n$. Now if $N=10^k$ with $k$ even then the probability of 0 is $\ge .9$, and if $k$ is odd, then the probability of $1$ is $\ge .9*1/3=.3$.  In general, the limit exists if the intervals between consecutive numbers in $A$ are "uniformly spaced". In particular, if $A$ is a set of primes, I do not know how to show that the limit exists. It may be a hard number theory problem. For example, we know (Green and Tao) that the set of primes contains arbitrary long arithmetic progressions, but it is not clear (to me) how often these occur and how often progressions start at relatively small numbers and are relatively long.