Two different numbers are randomly selected from the set $\{1, 2, 3, 4\}$ and they are multiplied. What is the probability that the product is even? Express your answer as a common fraction.
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 $\binom{4}{2}=6$ pairs, the probability of drawing $\{1,3\}$ is $\frac{1}{6}$.  So the probability of drawing two numbers whose product is even is $1-\dfrac{1}{6}=\boxed{\dfrac{5}{6}}$.