Let's start by calculating $P(A \cap B)$.

We know the formula:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Given:

• $P(A) = 0.3$
• $P(B) = 0.4$
• $P(A \cup B) = 0.67$

Substituting these values into the formula:

$0.67 = 0.3 + 0.4 - P(A \cap B)$

$P(A \cap B) = 0.3 + 0.4 - 0.67 = 0.7 - 0.67 = 0.03$

So, $P(A \cap B) = 0.03$.

Now, let's answer the questions:

(a) Are events $A$ and $B$ independent?

Events $A$ and $B$ are independent if:

$P(A \cap B) = P(A) \times P(B)$

Let's check:

$P(A) \times P(B) = 0.3 \times 0.4 = 0.12$

Since $P(A \cap B) = 0.03$ and $P(A) \times P(B) = 0.12$, $A$ and $B$ are not independent.

Answer: NO

(b) Are $A$ and $B$ mutually exclusive (disjoint)?

Events $A$ and $B$ are mutually exclusive if:

$P(A \cap B) = 0$

Since $P(A \cap B) = 0.03$, $A$ and $B$ are not mutually exclusive.

Answer: NO

Would you like any further details or have any questions? Here are some related questions you might find interesting:

1. How do you determine if two events are independent?
2. What does it mean for two events to be mutually exclusive?
3. Can two events be both independent and mutually exclusive?
4. How does the addition rule for probabilities work?
5. What is the relationship between conditional probability and independence?

Tip: Always double-check the definitions and conditions for independence and mutual exclusivity to avoid confusion in probability problems.