jmc

The product of four positive integers is prime only if three of the integers are 1 and the fourth is a prime number. Therefore, of the $6^4$ outcomes for the roll of four dice, only the outcomes $(1,1,1,p)$, $(1,1,p,1)$, $(1,p,1,1)$, and $(p,1,1,1)$ for $p=2$, $3$, or $5$ give a prime product. The probability of achieving a prime product is therefore $$\frac{12}{6\cdot 6\cdot 6\cdot 6}=\frac{2}{6\cdot 6\cdot 6}=\frac{1}{3\cdot 6^2}=\boxed{\frac{1}{108}}.$$