Question

A fair die is rolled 5 times. the probability of getting a number greater than 2 at least three times i

Answer 1

This amounts to asking for the probability of having exactly twice a number in the set {1, 2, 3} and exactly three times a number in the set {4, 5, 6} out of the 5 results you get.

Getting a number in the set A = {1, 2, 3} has probability 3/6, i.e., 1/2.

Getting a number in the set B = {4, 5, 6} has probability 3/6, i.e., 1/2 too.

So getting twice from A and thrice from B has probability (1/2)^5 = 1/32

Since there are 5-Choose-2 (5! / (3! . 2!) ) = 10 ways to pick which of the 2 throws are to come from set A and which of the remaining 3 throws are to come from set B, our answer for the probability is 10 / 32, i.e., 5/16.