if 0<=x<=1, then f(x) = sum_{i=0}^{[x]} (-1)^i C(n,i) (x-i)^(n-1) / (n-1)!

This has the similar idea to a Pascal Triangle.
Integrating f(x) from (0.5n+0.5) to n will give the answer. When n is big enough, f(x) is close to be symmetric, and the probability will be close to 0.5.