### Instruction: Six 6-sided dice are rolled. What is the probability that three of the dice show prime numbers and the rest show composite numbers?### Response: I know that the prime numbers on a 6-sided die are 2, 3, and 5, and the composite numbers are 1, 4, and 6.
So each die has a 1/2 chance of showing a prime number and a 1/2 chance of showing a composite number.
I want to find the probability of getting exactly three prime numbers and three composite numbers, regardless of the order.
This is a binomial probability problem, where the number of trials is 6, the number of successes is 3, and the probability of success is 1/2.
The binomial probability formula is P(X = k) = nCk * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the probability of success, and nCk is the number of ways to choose k items from n items.
In this case, n = 6, k = 3, and p = 1/2.
So I can plug these values into the formula and simplify.
P(X = 3) = 6C3 * (1/2)^3 * (1-1/2)^(6-3) = 20 * (1/8) * (1/8) = 20/64 = 5/16.
So the probability of getting three prime numbers and three composite numbers on six dice is 5/16.