Recently I come up with an embarrassingly easy question. It should be known or elementary but I am still not able to find either a correct answer or references:

"Consider a random vector $x=(x_1,...,x_n)$ in the simplex $0\le x_i, x_1+..+x_n=1$. It is easy to show that each $x_i$ has beta distribution $B(1,n-1)$. It can be also checked that the expected value of $|x|_2$ is of order $n^{-1/2}$.

I am wondering if there is any concentration result saying that there exists a sufficiently large constant $C$ such that $|x|_2 \le Cn^{-1/2}$ with high probability, say $1-n^{-3}$? "

Thanks.