Let $p_1,\ldots,p_s$ be $s$ number in the unit interval such that $p_1+\ldots+p_s=1.$
Is it then true, that for every $n\geq 1$ we have $ \sum_{(k_1,\ldots,k_n)\in \{1,\ldots,s \}^n} p_{k_1}\cdot \ldots \cdot p_{k_n}=1 ?$
Checking it for $n=2$ indicates that it's true (but already for $n=3$ it isn't feasable to check it by hand)...