jmc

The product of two integers is even unless both of the integers are odd. Therefore, the only pair of integers from the set $\{1,2,3,4\}$ whose product is odd is $\{1,3\}$. Since there are $\left(\genfrac{}{}{0ex}{}{4}{2}\right)=6$ pairs, the probability of drawing $\{1,3\}$ is $\frac{1}{6}$. So the probability of drawing two numbers whose product is even is $1-\frac{1}{6}=\boxed{\frac{5}{6}}$.