Let’s analyze and solve each part of the problem step by step.

Given Information

We have three events $A$, $B$, and $C$ with the following probabilities:

• $P(A^c) = 0.3$
• $P(B) = 0.5$
• $P(C) = 0.2$
• $P(B \cup C) = 0.6$
• $P(A \cap B) = 0.4$

We can derive $P(A)$ using the complement rule: $P(A) = 1 - P(A^c) = 1 - 0.3 = 0.7$

Part (a): Find $P(B \cap C)$

Using the formula for the union of two events: $P(B \cup C) = P(B) + P(C) - P(B \cap C)$ Substituting the known values: [ 0.6