If p(a) = 0.2, p(b) = 0.2, and p(a ∪ b) = 0.4, then p(a ∩ b) =
(a) are events a and b independent? (yes or no)
(b) are a and b mutually exclusive? (yes or no)

P(A ∩ B) = 0.

a) NO

b) YES

Step-by-step explanation:

Thinking about this through Venn diagrams we can sort of understand that:

if P(A) = 0.2 and P(B) = 0.2, and P(A∪B) = 0.4.

there's no overlapping between P(A) and P(B).

(If there was overlapping then P(A∪B) < 0.4, since you'd be excluding the overlapped part from getting counted twice.

Think of it in terms of calculating areas circles A and B, if the circles were disjoint, then the sum of the areas A and B would be 0.2+0.2. But if the circles were overlapping then the sum of the areas would be 0.2+0.2-P(A ∩ B), where P(A ∩ B) is the overlapping part)

since there's no overlapping P(A ∩ B) = 0.

a) NO

events A and B are only independent when P(A ∩ B) > 0 (or overlapping)

b) YES

events A and B are mutually exclusive when P(A ∩ B) = 0 (or disjoint)

step-by-step explanation: