I was reading a book on Monte Carlo methods and now I'm trying to make sense of an excercise. At one point they say that according to the central limit theorem most of the points ${\bf x} \in [0,1]^n$ are near the hyperplane $\{{\bf x} \mid \sum_i^n x_i = n/2\}$ which passes through the center of the hypercube. Then, if the distance of some point $\mathbf x$ to the hyperplane is $$d_h({\bf x}) = \frac{1}{\sqrt{n}} \bigg |\sum_i^n (x_i - 1/2) \bigg | $$ we can define a set $B_\epsilon = \{ {\bf x} \in [0,1]^n \mid d_c({\mathbf x}) \leq \epsilon \}$ which has volume nearly $1 - 2\Phi(-\epsilon\sqrt{12n})$, for large $n$.
So, what does $\Phi$ stand for in this statement? Is it the PDF of the standard normal distribution? Also, I would like to know how this approximation of the volume is derived and how is this related to the central limit theorem.