solve for elements of the sample space

Question

Suppose that for three events A, B, C in some sample space S, we know the following:
• A ⊆ B (i.e. A is a subset of B)
• P(A) = 0.3
• P(B) = 0.5
• P(C) = 0.55
• P(A ∩ C) = 0.1
• P(Ac ∩ Bc ∩ Cc) = 0.1

(a) P(A ∩ B)
(b) P(B ∩ C)
(c) P(A ∪ C)
(d) P(C|B)
(e) Are the events A and C independent? Why or why not?

Answer 1

Given information: A ⊆ B (i.e. A is a subset of B) and

• P(A) = 0.3

• P(B) = 0.5

• P(C) = 0.55

• P(A ∩ C) = 0.1

• P(A^c ∩ B^c ∩ C^c) = 0.1

a.) P(A ∩ B) = P(A)*P(B) = (0.3)(0.5) = 0.15

b.) P(B ∩ C) = P(B)*P(C) = (0.5)(0.55) = 0.275

c.) P(A ∪ C) = P(A) + P(C) - P(A ∩ C) = 0.3 + 0.55 - 0.1 = 0.75

d.) P(C | B) = P(B ∩ C) / P(B) = 0.275 / 0.5 = 0.55

e.) P(A ∩ C) = P(A)*P(C) = (0.3)(0.55) = 0.165 ≠ 0.1

Events A and C are dependent because one event is affecting the outcome of the other event.