To find \( P(27 \cap (28 \cup 29)) \), we need to apply some basic rules from probability theory, particularly focusing on the intersection (\( \cap \)) and union (\( \cup \)) of events.

1. **Union Rule**:  
   The probability of the union of two events \( A \) and \( B \) is given by:
   \[
   P(A \cup B) = P(A) + P(B) - P(A \cap B)
   \]
   So for \( 28 \cup 29 \), we can write:
   \[
   P(28 \cup 29) = P(28) + P(29) - P(28 \cap 29)
   \]

2. **Intersection Rule**:  
   The probability of the intersection of event \( 27 \) with the union of \( 28 \cup 29 \) is:
   \[
   P(27 \cap (28 \cup 29)) = P((27 \cap 28) \cup (27 \cap 29))
   \]
   And this can be further expanded as:
   \[
   P(27 \cap (28 \cup 29)) = P(27 \cap 28) + P(27 \cap 29) - P(27 \cap 28 \cap 29)
   \]

To calculate this exactly, we need the individual probabilities and intersections between the events \( 27 \), \( 28 \), and \( 29 \). Without those values, we can only express the final probability in terms of these components.

Would you like to provide those probabilities, or should we continue with this structure? Feel free to ask any follow-up questions!

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**Relative Questions:**

1. What are the individual probabilities of events 27, 28, and 29?
2. How do we calculate the probability of the intersection between two events?
3. What happens if events 27, 28, and 29 are mutually exclusive?
4. How does the inclusion-exclusion principle apply to three events?
5. Can we simplify the expression if the events are independent?

**Tip:** When dealing with probabilities of intersections and unions, always check if events are independent or mutually exclusive—it simplifies calculations greatly!

Explore how to solve the probability problem P(27∩(28∪29)) using the inclusion-exclusion principle. This problem demonstrates key probability concepts such as union, intersection, and the formula for calculating the probability of combined events. Suitable for students in Grades 10-12.

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