Probability of getting non-negative real number n-tuple (which sum=S) in which every variable is not greater than S/2

Question

n non-negative real numbers choosen such that their sum=S . What is probability of getting n-tuple such that every number of tuple is less than or equal to S/2?

Answer 1

If you draw a point uniformly (with respect to the $(n-1)$-dimensional Lebesgue measure) from the set $\{\,(x_1,\ldots,x_n)\mid x_i\ge 0,\sum x_i=S\,\}$, then by a simple scaling argument, $P(x_n>a)$ is proportional to $(S-a)^{n-1}$. From $P(x_n>0)=1$, we conclude $$ P(x_n>a)=\left(1-\frac aS\right)^{n-1}$$ and in particular $$ P(x_n>\tfrac S2)=2^{1-n}.$$ The same holds with $x_i$ in place of $x_n$. As the events $x_i>\frac S2$ are mutually exclusive, we conclude $$ P(\forall i\colon x_i\le\tfrac S2)=1-P(\exists i\colon x_i>\tfrac S2)=1-\sum_iP(x_i>\tfrac S2)=1-n2^{1-n}$$