SCHWEITZER 2004 Problem 5

Let $G$ be a non-solvable finite group and let $\varepsilon > 0$ . Show that there exist a positive integer $k$ and a word $w\in F_k$ such that $w$ assumes the value $1$ with probability less than $\varepsilon$ when its $k$ arguments are considered to be independent and uniformly distributed random variables with values in $G$ . (We write $F_k$ for the free group generated by $k$ elements.)