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We are tasked with finding \( P(A \cup \overline{B}) \), given the following probabilities:

- \( P(A) = 0.9 \)
- \( P(B) = 0.7 \)
- \( P(A \cap B) = 0.6 \)

### Step-by-step solution:

We know the formula for the probability of a union:

\[
P(A \cup \overline{B}) = P(A) + P(\overline{B}) - P(A \cap \overline{B})
\]

1. **Calculate \( P(\overline{B}) \):**
   \[
   P(\overline{B}) = 1 - P(B) = 1 - 0.7 = 0.3
   \]

2. **Calculate \( P(A \cap \overline{B}) \):**
   Using the fact that \( P(A \cap B) \) is given, we can calculate \( P(A \cap \overline{B}) \) using the complement rule:
   \[
   P(A \cap \overline{B}) = P(A) - P(A \cap B) = 0.9 - 0.6 = 0.3
   \]

3. **Substitute into the union formula:**
   \[
   P(A \cup \overline{B}) = P(A) + P(\overline{B}) - P(A \cap \overline{B})
   \]
   \[
   P(A \cup \overline{B}) = 0.9 + 0.3 - 0.3 = 0.9
   \]

Thus, the correct answer is \( P(A \cup \overline{B}) = 0.9 \), which corresponds to option **d**.

---

Would you like any additional explanation or have further questions?

Here are some related questions for deeper understanding:
1. What is the complement of an event in probability theory?
2. How do you find the intersection of two events?
3. Can the probability of a union exceed 1? Why or why not?
4. How does the complement rule apply to conditional probability?
5. What is the difference between independent and mutually exclusive events in probability?

**Tip**: Always double-check the formula you're using, especially when dealing with intersections and unions involving complements.

If P(A) = 0.9, P(B) = 0.7, P(A ∩ B) = 0.6, then find P(A ∪ B').

Solve the probability problem P(A ∪ B') given P(A) = 0.9, P(B) = 0.7, and P(A ∩ B) = 0.6 using key concepts such as union, complement, and intersection.

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