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