ASV Intro to Probability 1.20

Question

We roll a fair die four times.

(a) Describe the sample space Ω and the probability measure P that model this

experiment. To describe P , give the value P (ω) for each outcome ω ∈ Ω.

(b) Let A be the event that there are at least two fives among the four rolls. Let B

be the event that there are at most one five among the four rolls. Find the probabilities

P (A) and P (B) by finding the ratio of the number of favorable outcomes to the total.

(c) What is the set A ∪ B? What equality should P (A) and P (B) satisfy? Check

that your answers to part (b) satisfy this equality.

Answer 1

(a) Ω = { (a, b, c, d), a, b, c, d ∈ {1, 2, 3, 4, 5, 6} }

P(ω) = (1/6)⁴, for all ω ∈ Ω

(b) P(zero 5's) = 4choose0 ×(5/6)⁴ = (5/6)⁴

P(only one 5) = 4choose1 × (1/6)(5/6)³ = 4 × (1/6)(5/6)³

P(A) = 1 - (P(zero 5's) + P(only one 5))

P(B) = P(zero 5's) + P(only one 5)

(c) A υ B = Ω because A and B are complements of each other. So P(A) + P(B) = 1