If an unbiased die, marked with - 2, - 1, 0, 1, 2, 3 on its faces...

The unbiased die is thrown five times such that the product of the outcomes is positive, then either all outcomes are positive or any two are negative.

Either all outcomes are positive or any two are negative.

Now,

$p = p$ (positive) $= \dfrac{3}{6} = \dfrac{1}{2}$

$q = p$ (negative) $= \dfrac{2}{6} = \dfrac{1}{3}$

Required probability

$= C5 5 {\left(\dfrac{1}{2}\right)}^{5} + C2 5 {\left(\dfrac{1}{3}\right)}^{2}{\left(\dfrac{1}{2}\right)}^{3} + C4 5 {\left(\dfrac{1}{3}\right)}^{4}{\left(\dfrac{1}{2}\right)}^{1}$

$= \dfrac{521}{2592}$