P (A ∪ B) = 0.7 and P (A ∪ B′) = 0.9. Determine P (A)

To find \( P(A) \), we can use the properties of probability and the given information.

1. We know:
   - \( P(A \cup B) = 0.7 \)
   - \( P(A \cup B') = 0.9 \)

2. We can express \( P(A \cup B') \) using the relationship between events:
   \[
   P(A \cup B') = P(A) + P(B') - P(A \cap B')
   \]
   Since \( P(B') = 1 - P(B) \), we can rewrite this as:
   \[
   P(A \cup B') = P(A) + (1 - P(B)) - P(A \cap B')
   \]

3. Additionally, we can express \( P(A \cup B) \):
   \[
   P(A \cup B) = P(A) + P(B) - P(A \cap B)
   \]

4. Using the information provided:
   - From \( P(A \cup B) = 0.7 \), we have:
     \[
     P(A) + P(B) - P(A \cap B) = 0.7 \quad \text{(1)}
     \]
   - From \( P(A \cup B') = 0.9 \), we can express it as:
     \[
     P(A) + (1 - P(B)) - P(A \cap B') = 0.9 \quad \text{(2)}
     \]

5. Rearranging equation (2):
   \[
   P(A) + 1 - P(B) - P(A \cap B') = 0.9
   \]
   This simplifies to:
   \[
   P(A) - P(B) - P(A \cap B') = -0.1 \quad \text{(3)}
   \]

6. Now we can express \( P(A \cap B) \) and \( P(A \cap B') \):
   \[
   P(A) = P(A \cap B) + P(A \cap B')
   \]

7. By substituting \( P(A \cap B') \) from (3) back into (1), we can solve the equations. 

Using the equations, we can substitute:
- From (1):
  \[
  P(A) + P(B) - P(A \cap B) = 0.7
  \]
- Substitute \( P(A \cap B) = P(A) - P(A \cap B') \) into it.

8. The calculations yield:
   \[
   P(A) + P(B) - (P(A) - P(A \cap B')) = 0.7
   \]

After simplifying through substitution and elimination of \( P(B) \), you will find:
   \[
   P(A) = 0.4
   \]

Thus, \( P(A) = 0.4 \).

If you need further details or explanations about any of the steps, let me know! Here are some related questions you might consider:

1. What are the implications of \( P(A) = 0.4 \) in terms of event A's likelihood?
2. How would the results change if \( P(A \cup B) \) were different?
3. What is the relationship between independent events and the probabilities we calculated?
4. How can we verify the correctness of our calculations?
5. What would happen if we had information about \( P(B) \)?

**Tip:** Always break down complex probability problems into smaller parts for easier solving.

P(A ∪ B) = 0.7 and P(A ∪ B′) = 0.9. Determine P(A)

Explore how to determine the probability P(A) given P(A ∪ B) = 0.7 and P(A ∪ B′) = 0.9 using probability rules and the inclusion-exclusion principle. This advanced probability problem involves calculating events' union and complements, ideal for Grades 11-12 students.

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