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**Problem Statement:**
- Consider two events \( C \) and \( D \) such that \( P(C) = 0.4 \), \( P(D) = 0.6 \), and \( P(C \cap D) = 0.25 \).
- a) Draw a Venn diagram showing the probabilities for events \( C \) and \( D \).

**Solution:**

To draw a Venn diagram showing the probabilities, we need to understand the relationships between the given probabilities:

1. \( P(C) = 0.4 \) is the probability of event \( C \) occurring.
2. \( P(D) = 0.6 \) is the probability of event \( D \) occurring.
3. \( P(C \cap D) = 0.25 \) is the probability of both events \( C \) and \( D \) occurring together (the intersection of \( C \) and \( D \)).

Using this information, we can calculate the probabilities of the other regions:

- **Only \( C \)**: \( P(C \text{ only}) = P(C) - P(C \cap D) = 0.4 - 0.25 = 0.15 \)
- **Only \( D \)**: \( P(D \text{ only}) = P(D) - P(C \cap D) = 0.6 - 0.25 = 0.35 \)
- **Neither \( C \) nor \( D \)**: This is calculated as \( 1 - P(C \cup D) \), where \( P(C \cup D) \) is the probability of either \( C \) or \( D \) or both occurring.
  
   \( P(C \cup D) = P(C) + P(D) - P(C \cap D) = 0.4 + 0.6 - 0.25 = 0.75 \)
  
   So, \( P(\text{neither } C \text{ nor } D) = 1 - 0.75 = 0.25 \).

Thus, the Venn diagram would show:
- The intersection \( C \cap D \) with a probability of \( 0.25 \).
- The region corresponding to only \( C \) with a probability of \( 0.15 \).
- The region corresponding to only \( D \) with a probability of \( 0.35 \).
- The region outside both circles representing neither \( C \) nor \( D \) with a probability of \( 0.25 \).

Would you like an illustration of this Venn diagram, or do you need further details on any part of this explanation?

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### Relative Questions:
1. What is the general formula for calculating \( P(C \cup D) \)?
2. How would you calculate \( P(C \cap D) \) if only \( P(C) \) and \( P(D) \) were known, assuming independence?
3. What is the significance of the union and intersection in probability theory?
4. How would the Venn diagram change if \( C \) and \( D \) were mutually exclusive?
5. How does the addition rule for probability apply when events are not mutually exclusive?

### Tip:
When dealing with probabilities, always ensure that the sum of all disjoint events (including their complements) adds up to 1. This consistency check can help you avoid mistakes.

Consider two events C and D such that P(C) = 0.4, P(D) = 0.6, and P(C ∩ D) = 0.25. a) Draw a Venn diagram showing the probabilities for events C and D.

Learn how to draw a Venn diagram to represent the probabilities of two events C and D, with P(C) = 0.4, P(D) = 0.6, and their intersection P(C ∩ D) = 0.25. This problem explores probability concepts, including the addition rule, and is suitable for students in Grades 9-12.

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