The sum of independent identically distributed random variables follows the Irwin-Hall distribution. For $n$ variables that are uniformly distributed on (0,1), the probability density function is $$f_X(x)=\frac{1}{2\left(n-1\right)!}\sum_{k=0}^{n}\left(-1\right)^k{n \choose k}\left(x-k\right)^{n-1}\text{sgn}(x-k).$$
So for $n=11$ you will have to integrate an explicit piecewise polynomial function.
