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.