To solve this problem, we need to find $P(A \cap B)$ and $P(A' \cup B)$, given the following information:

1. $P(A) = 0.2$
2. $P(B) = 0.4$
3. $P(A \cup B) = 0.5$

Let's solve each part step-by-step:

Part (a): Finding $P(A \cap B)$

We use the formula for the union of two events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Substitute the given values into the formula: $0.5 = 0.2 + 0.4 - P(A \cap B)$

Simplify and solve for $P(A \cap B)$: $0.5 = 0.6 - P(A \cap B)$ $P(A \cap B) = 0.6 - 0.5$ $P(A \cap B) = 0.1$

Part (b): Finding $P(A' \cup B)$

We use the formula for the complement of $A$: $P(A') = 1 - P(A)$ Substitute the given value for $P(A)$: $P(A') = 1 - 0.2 = 0.8$

Now, we need to find $P(A' \cup B)$, which can be calculated using the formula for the union of two events: $P(A' \cup B) = P(A') + P(B) - P(A' \cap B)$

To find $P(A' \cap B)$, we use the fact that $P(A' \cap B) = P(B) - P(A \cap B)$ (since $A' \cap B$ represents the part of $B$ that does not overlap with $A$): $P(A' \cap B) = 0.4 - 0.1 = 0.3$

Now, substitute back into the formula for $P(A' \cup B)$: $P(A' \cup B) = 0.8 + 0.4 - 0.3$ $P(A' \cup B) = 1.2 - 0.3 = 0.9$

Final Answers:

• (a) $P(A \cap B) = 0.1$
• (b) $P(A' \cup B) = 0.9$

Would you like any additional explanations or details on these calculations?

Here are some related questions to explore further:

1. How would you find $P(A \cup B')$ given the same information?
2. What is the probability of $P(A' \cap B')$?
3. How would these probabilities change if $A$ and $B$ were independent events?
4. Can you use Venn diagrams to illustrate these probabilities?
5. How would you find $P(A | B)$ and $P(B | A)$ with this information?

Tip: Always verify the consistency of probability values to ensure that the solution is correct, especially when dealing with complements and intersections.