1. [15] Four standard six-sided dice are rolled. Find the probability that, for each pair of dice, the product of the two numbers rolled on those dice is a multiple of 4 .
Answer: $\square$
If any two of the dice show an odd number, then this is impossible, so at most one of the dice can show an odd number. We take two cases:
Case 1: If exactly one of the dice shows an odd number, then all three other dice must show a multiple of 4 , which can only be the number 4 . The probability that this occurs is $4 \cdot \frac{1}{2} \cdot\left(\frac{1}{6}\right)^{3}=\frac{1}{108}$.
Case 2: If all of the dice show even numbers, then the condition is satisfied. The probability that this occurs is $\left(\frac{1}{2}\right)^{4}=\frac{1}{16}$.
The total probability is $\frac{1}{108}+\frac{1}{16}=\frac{31}{432}$.