Question

Suppose you are rolling a die for 4 times.

Let x be the random variable finding how many times you will get a ‘3’.

a. List all possible values for x.

b. Find the probability distribution of x.

Answer 1

a. Possible values for X = 0,1,2,3,4

b. The sample space of equally likely outcomes is

$P(3) = \frac{34}{36} \\ p=0.055 \\ q = 0.945 \\ n = 4 \\ P(X=0) = C^4_0 (0.055)^0(0.945)^{4-0} \\ = \frac{4!}{0!(4-0)!} \times 1 \times 0.7974 \\ = 0.7974 \\ P(X=1) = C^4_1 (0.055)^1(0.945)^{4-1} \\ = \frac{4!}{1!(4-1)!} \times 0.055 \times 0.8439 \\ = 0.1856 \\ P(X=2) = C^4_2 (0.055)^2(0.945)^{4-2} \\ = \frac{4!}{2!(4-2)!} \times 0.003025 \times 0.8930 \\ = 0.0162 \\ P(X=3) = C^4_3 (0.055)^3(0.945)^{4-3} \\ = \frac{4!}{3!(4-3)!} \times 0.00016638 \times 0.945 \\ = 0.0006289 \\ P(X=4) = C^4_4 (0.055)^4(0.945)^{4-4} \\ = \frac{4!}{4!(4-4)!} \times 9.15 \times 10^{-6} \times 1 \\ = 9.15 \times 10^{-6}$