(1) What is Pr[B] if A and B are independent events? (2) What is Pr[B] if A and B are disjoint events?

Question

For this problem, assume that Pr[A ∪B] = 0.6 and Pr[A] = 0.35.
(1) What is Pr[B] if A and B are independent events?

(2) What is Pr[B] if A and B are disjoint events?

Answer 1

For any events A and B, we have:

Equation 1: Pr[A ∪ B] = Pr[A] + Pr[B] - Pr[A ∩ B].

Let's do Part (2) first, as this is easier:

Part (2) If A and B are disjoint events, then P[A ∩ B] = 0. We are given Pr[A ∪ B] = 0.6 and Pr[A] = 0.35. Plugging all of these into Equation 1 give us

0.6 = 0.35 + Pr[B] + 0

So, Pr[B] = 0.6 - 0.35 = 0.25.

Part (1). A and B are independent means the following:

Equation 2: Pr[A ∩ B] = Pr[A] · Pr[B]

Therefore, Pr[A ∩ B] = 0.35·Pr[B]. Plugging this and the information given in the problem into Equation 1 yields:

Equation 3: 0.6 = 0.35 + Pr[B] - 0.35·Pr[B]

Note: Pr[B] - 0.35·Pr[B] = 1.0·Pr[B] - 0.35·Pr[B] = Pr[B]·(1.0 - 0.35) = 0.65·Pr[B]

Therefore, from Equation 3, we have 0.6 = 0.35 + 0.65·Pr[B], and so

Pr[B] = (0.6 - 0.35) / 0.65 = 0.3846... = 5/13.