Question

six $6$-sided dice are rolled. what is the probability that three of the dice show prime numbers and the rest show composite numbers?

Answer 1

The probability of rolling three prime numbers and three composite numbers on six 6-sided dice can be calculated using binomial coefficients.

Step-by-step explanation:

In order to find the probability of rolling three prime numbers and three composite numbers on six 6-sided dice, we need to consider the total number of outcomes and the number of favorable outcomes.

There are a total of 6^6 = 46656 possible outcomes when rolling six 6-sided dice.

The number of favorable outcomes is found by calculating the product of two binomial coefficients. The first binomial coefficient represents the number of ways to choose 3 dice out of 6 that show prime numbers, which is denoted as C(6,3). The second binomial coefficient represents the number of ways to choose 3 dice out of 6 that show composite numbers, which is also denoted as C(6,3).

Therefore, the probability of rolling three prime numbers and three composite numbers on six 6-sided dice is:

P = (C(6,3) * C(6,3)) / 6^6.

Answer 2

The probability of rolling three primes and three composites on six 6-sided dice is 25 / 2916.

Step-by-step explanation:

To find the probability of rolling three primes and three composites on six 6-sided dice, we need to determine the number of favorable outcomes and the total number of possible outcomes.

1. Calculate the number of ways to roll three primes on six dice. There are three prime numbers on a 6-sided die: 2, 3, and 5. So, the number of ways to choose three primes from six is given by the combination formula: C(6, 3) = 6! / (3!(6-3)!) = 20.

2. Calculate the number of ways to roll three composites on six dice. There are three composite numbers on a 6-sided die: 4, 6. So, the number of ways to choose three composites from six is also 20.

3. Calculate the total number of possible outcomes when rolling six dice. On each die, there are six possible outcomes, so the total number of outcomes is 6^6 = 46656.

4. Finally, calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes: P = (20 * 20) / 46656 = 400 / 46656 = 25 / 2916.