HMMT 2015 U35

Let $P$ denote the set of all subsets of $\{1, \ldots, 23\}$. A subset $S \subseteq P$ is called good if whenever $A, B$ are sets in $S$, the set $(A \backslash B) \cup(B \backslash A)$ is also in $S$. (Here, $A \backslash B$ denotes the set of all elements in $A$ that are not in $B$, and $B \backslash A$ denotes the set of all elements in $B$ that are not in $A$.) What fraction of the good subsets of $P$ have between 2015 and 3015 elements, inclusive? If your answer is a decimal number or a fraction (of the form $m / n$, where $m$ and $n$ are positive integers), then your score on this problem will be equal to $\max \{0,25-\lfloor 1000|A-N|\rfloor\}$, where $N$ is your answer and $A$ is the actual answer. Otherwise, your score will be zero.