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To solve this problem, we need to find \( P(B|A) \), which is the conditional probability of \( B \) given \( A \).

### Given:
- \( P(A) = \frac{3}{5} \)
- \( P(A \cup B) = \frac{2}{3} \)
- \( A \) and \( B \) are independent events.

### Steps to Solve:

1. **Use the formula for the union of two events:**

   \[
   P(A \cup B) = P(A) + P(B) - P(A \cap B)
   \]
   
   Since \( A \) and \( B \) are independent:
   
   \[
   P(A \cap B) = P(A) \cdot P(B)
   \]
   
   Substituting this into the union formula gives:
   
   \[
   P(A \cup B) = P(A) + P(B) - P(A) \cdot P(B)
   \]

2. **Substitute the known values:**

   \[
   \frac{2}{3} = \frac{3}{5} + P(B) - \left(\frac{3}{5} \cdot P(B)\right)
   \]
   
   Simplify the equation:
   
   \[
   \frac{2}{3} = \frac{3}{5} + P(B) \left(1 - \frac{3}{5}\right)
   \]
   
   \[
   \frac{2}{3} = \frac{3}{5} + P(B) \cdot \frac{2}{5}
   \]
   
   Multiply everything by 15 to eliminate the denominators:
   
   \[
   10 = 9 + 6P(B)
   \]
   
   Subtract 9 from both sides:
   
   \[
   1 = 6P(B)
   \]
   
   Therefore, \( P(B) = \frac{1}{6} \).

3. **Find \( P(B|A) \):**

   \[
   P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{P(A) \cdot P(B)}{P(A)} = P(B)
   \]
   
   Substituting the value of \( P(B) \):
   
   \[
   P(B|A) = \frac{1}{6}
   \]

### Final Answer:
\[
P(B|A) = \frac{1}{6}
\]

Would you like more details, or do you have any further questions? 

Here are five related questions for practice:

1. If \( P(A) = 0.7 \) and \( P(B) = 0.4 \), and \( A \) and \( B \) are independent, what is \( P(A \cap B) \)?
2. Given that \( P(A) = 0.5 \), \( P(B) = 0.3 \), and \( P(A \cup B) = 0.65 \), find \( P(A \cap B) \).
3. If \( P(A \cup B) = 0.9 \) and \( P(A \cap B) = 0.2 \), what is \( P(A) \) if \( P(B) = 0.4 \)?
4. What is the conditional probability \( P(A|B) \) if \( P(B) = 0.2 \) and \( P(A \cap B) = 0.1 \)?
5. How would the independence of events \( A \) and \( B \) affect the calculation of \( P(A \cup B) \)?

**Tip:** When dealing with conditional probabilities, remember that the definition \( P(B|A) = \frac{P(A \cap B)}{P(A)} \) can simplify many problems, especially when events are independent.

Events A and B are independent. P(A) = 3/5 and P(A ∪ B) = 2/3. Find P(B|A). Give an exact answer as a fraction in fully simplified form.

Learn how to find the conditional probability P(B|A) when given P(A) = 3/5 and P(A ∪ B) = 2/3, with A and B being independent events. This problem involves key concepts in probability, including the union and conditional probability formulas. Suitable for students in Grades 10-12.

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