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### Solutions:

1. **Why is \( P(A \cap B) = 0 \) for mutually exclusive events?**

   Two events are mutually exclusive if they cannot occur simultaneously. This means that the intersection of the two events, \( A \cap B \), contains no outcomes. Consequently, the probability of both events happening together, \( P(A \cap B) \), is zero.

2. **Examples:**
   - **(a) Two events that are mutually exclusive:**
     - Event \( A \): Rolling a 1 on a six-sided die.
     - Event \( B \): Rolling a 2 on the same die.
     - These events are mutually exclusive because a single die roll cannot result in both a 1 and a 2 at the same time.
   - **(b) Two events that are not mutually exclusive:**
     - Event \( C \): Rolling an even number on a six-sided die (\{2, 4, 6\}).
     - Event \( D \): Rolling a number greater than 3 on the same die (\{4, 5, 6\}).
     - These events are not mutually exclusive because they can occur together (e.g., rolling a 4 or 6 satisfies both conditions).

Would you like a more detailed breakdown or further examples? Here are five related questions to deepen your understanding:

1. What is the mathematical formula for determining the probability of the union of two mutually exclusive events?
2. Can two events with non-zero intersection be mutually exclusive? Why or why not?
3. How do mutually exclusive events differ from independent events in probability?
4. What is the formula for the union probability \( P(A \cup B) \) for events that are not mutually exclusive?
5. Can an event be mutually exclusive with itself? Why or why not?

**Tip:** When analyzing probabilities, always carefully check whether events overlap in their outcomes—this distinction defines whether they are mutually exclusive.

1. When two events are mutually exclusive, why is P(A and B) = 0? 2. Give an example of (a) two events that are mutually exclusive and (b) two events that are not mutually exclusive.

Explore why P(A and B) = 0 for mutually exclusive events with examples. Learn the key concepts of probability and distinguish between mutually exclusive and overlapping events. Suitable for high school students in Grades 9-12.

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