Six 6-sided dice are rolled. What is the probability that three of the dice show prime numbers and the rest show composite numbers?
There are $\binom{6}{3}=20$ ways to choose three of the dice to show prime numbers. Each roll is prime with probability $\frac{1}{2}$ and composite with probability $\frac{1}{3}$, so each arrangement of 3 prime numbers and 3 composite numbers occurs with probability $\left(\frac{1}{2}\right)^{\!3}\left(\frac{1}{3}\right)^{\!3}.$ Therefore, the probability that three dice show prime numbers and the rest show composite numbers is $$20\cdot \left(\frac{1}{2}\right)^{\!3}\left(\frac{1}{3}\right)^{\!3}=\boxed{\frac{5}{54}}.$$