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Let's break down the given problem by addressing each part in detail:

### a) Are \( B \) and \( G \) mutually exclusive events?

- **\( B \)** is the event that a randomly chosen person has **brown hair**.
- **\( G \)** is the event that a randomly chosen person has **green eyes**.

In the Venn diagram, we see the intersection of the two events, meaning there are people who have **both brown hair and green eyes** (e.g., Joan). Therefore, **\( B \) and \( G \) are *not* mutually exclusive**, since a person can be part of both events.

**Explanation**: Two events are mutually exclusive if they cannot occur at the same time. Since Joan has both brown hair and green eyes, \( B \) and \( G \) are not mutually exclusive.

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### b) Is Vanda correct in saying the probability of the person having brown hair, green eyes, or both is \( P(B) + P(G) \)?

Vanda's claim can be mathematically evaluated. If two events are **not** mutually exclusive, as in this case, the correct formula to compute the probability of either event happening (the union of the two events) is:

\[
P(B \cup G) = P(B) + P(G) - P(B \cap G)
\]

Where:
- \( P(B) \) is the probability of brown hair.
- \( P(G) \) is the probability of green eyes.
- \( P(B \cap G) \) is the probability of both brown hair and green eyes (the intersection).

Since Joan has both brown hair and green eyes, we must subtract the overlap \( P(B \cap G) \). Vanda incorrectly ignores the intersection, so **her claim is incorrect**.

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### Summary of Answers:
a) \( B \) and \( G \) are **not** mutually exclusive because some people (e.g., Joan) have both brown hair and green eyes.

b) Vanda is **incorrect** because when events are not mutually exclusive, you need to subtract the probability of both occurring. The correct formula is \( P(B \cup G) = P(B) + P(G) - P(B \cap G) \).

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Would you like to explore these concepts further or ask any specific questions? 

Here are 5 follow-up questions for better understanding:
1. What is the significance of mutually exclusive events in probability?
2. How can we calculate \( P(B) \), \( P(G) \), and \( P(B \cap G) \) from the diagram?
3. Can the probability of an event be greater than 1? Why or why not?
4. What happens if two events are independent instead of mutually exclusive?
5. How would the situation change if the events \( B \) and \( G \) were truly mutually exclusive?

**Tip**: Always remember to check if events are mutually exclusive or overlapping before calculating combined probabilities!

Are events B (brown hair) and G (green eyes) mutually exclusive? Vanda thinks the probability of the person having brown hair, green eyes, or both is P(B) + P(G). Is she correct?

Analyze probability in a Venn diagram involving brown hair and green eyes. Explore mutually exclusive events and compute the correct probability formula using P(B ∪ G) = P(B) + P(G) - P(B ∩ G).

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