Question

A fair 4-sided die is rolled 7 times.

(a)	Find the probability that the side 1 comes up exactly 3 times.

(b)	Find the probability that there is at least one side that comes up exactly 3 times.

Answer 1

Binomial distribution: P(X) = nCx p^x q^n-x

a) P(side 1), p = 1/4

q = 1 - 1/4 = 3/4

Number of rolls, n = 7

a) P(the side 1 comes up exactly 3 times) = 7C3 x (1/4)³ x (3/4)⁴

= 0.1038

b) P(at least one side that comes up exactly 3 times) = P(1 comes up exactly 3 times) + P(2 comes up exactly 3 times) + P(3 comes up exactly 3 times) + P(4 comes up exactly 3 times) - Number of ways to select 2 numbers from 4 x P(2 numbers comes up exactly 3 times)

= 0.1038x4 - 4C2 x (7C3 x 4C3) x (1/4)³ x (1/4)³ x (2/4)

= 0.4152 - 6x0.0171

= 0.3127