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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

Three events A, B, and C satisfy P(A^c) = 0.3, P(B) = 0.5, P(C) = 0.2, P(B ∪ C) = 0.6, P(A ∩ B) = 0.4. (a) What is P(B ∩ C)? (b) What is P(B^c | A)? (c) Are A and B independent?

Solve a probability problem involving three events A, B, and C. Calculate P(B ∩ C), conditional probability P(B^c | A), and check for independence between A and B using key probability concepts and formulas. Suitable for high school students in Grades 11-12.

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