6) Two coins are tossed. If A is the event "two heads" and B is the event "two tails” are A and B mutually exclusive? Are they complements?

Let's break this down by addressing the two key questions: whether events A and B are mutually exclusive, and whether they are complements.

### Definitions:
- **Event A**: Two heads (HH) appear.
- **Event B**: Two tails (TT) appear.

#### 1. **Are A and B mutually exclusive?**
- **Mutually exclusive events** are events that cannot occur simultaneously. If one event happens, the other cannot happen.
  
  In this case:
  - If two heads (HH) occur, then two tails (TT) cannot occur, and vice versa.
  - Thus, A and B cannot happen together.

  **Conclusion**: Yes, A and B are mutually exclusive.

#### 2. **Are A and B complements?**
- **Complementary events** are events where the occurrence of one means the other does not occur, and together, they represent all possible outcomes.

  In this scenario, the possible outcomes when tossing two coins are:
  - HH (Two heads)
  - TT (Two tails)
  - HT (One head, one tail)
  - TH (One head, one tail)
  
  Since the complement of an event must include all outcomes where that event does not occur, A (two heads) and B (two tails) are **not complements**. The outcomes HT and TH are not accounted for in either event.

  **Conclusion**: No, A and B are not complements.

### Final Answer:
- **Yes**, A and B are mutually exclusive.
- **No**, A and B are not complements.

Would you like further details or clarification?

Here are some related questions to explore:
1. What are some examples of events that are complements?
2. How would you calculate the probability of mutually exclusive events?
3. What is the probability of neither A nor B occurring?
4. How does the concept of mutually exclusive events relate to independent events?
5. Can an event be both mutually exclusive and independent?

**Tip**: Mutually exclusive events have no overlap, while complements cover all possible outcomes together!

Two coins are tossed. If A is the event 'two heads' and B is the event 'two tails,' are A and B mutually exclusive? Are they complements?

Explore the concept of mutually exclusive and complementary events with this probability problem involving two coin tosses. Understand why the events 'two heads' and 'two tails' are mutually exclusive but not complements, with clear explanations. Suitable for Grades 9-10.

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