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Let's solve the problem step by step:

We are given:
- \( P(A) = 0.2 \)
- \( P(B) = 0.4 \)
- \( P(A \cup B) = 0.5 \)

### Part (a) Find \( P(A \cap B) \)

We know from the formula of probability for union of two events:

\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]

Substitute the values:

\[
0.5 = 0.2 + 0.4 - P(A \cap B)
\]

Simplify:

\[
0.5 = 0.6 - P(A \cap B)
\]

Solving for \( P(A \cap B) \):

\[
P(A \cap B) = 0.6 - 0.5 = 0.1
\]

So, \( P(A \cap B) = 0.1 \).

---

### Part (b) Find \( P(A' \cup B) \)

\( A' \) represents the complement of \( A \). The formula for the probability of \( A' \cup B \) is:

\[
P(A' \cup B) = 1 - P(A \cap B')
\]

But we can also rewrite this using:

\[
P(A' \cup B) = 1 - P(A)
\]

This is because:

\[
P(A' \cup B) = P(B) + P(A')
\]

Now substitute values to find \( P(A' \cup B): and

If P(A) = 0.2, P(B) = 0.4, and P(A ∪ B) = 0.5, find (a) P(A ∩ B) and (b) P(A' ∪ B).

Learn how to solve probability problems involving union, intersection, and complementary events. This example calculates P(A ∩ B) and P(A' ∪ B) given P(A), P(B), and P(A ∪ B). Suitable for students in Grades 9-12.

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