Exactly three primes in six die rolls

A fair die is rolled six times, and a roll counts as a success when it shows a prime number. What is the probability of exactly three successes?

$\tfrac{5}{16}$. Three faces ($2,3,5$) are prime, so each roll succeeds with probability $\tfrac12$ and the count is $\text{Bin}(6,\tfrac12)$. Thus $\mathbb{P}=\binom63/2^6=\tfrac{20}{64}=\tfrac5{16}$.