If `P(A)=0.35,P(A uu B)=0.60`, find P(B), where A and B are independent events.

Answer

To solve the problem, we need to find \( P(B) \) given that \( P(A) = 0.35 \) and \( P(A \cup B) = 0.60 \), with the condition that events A and B are independent.

### Step-by-Step Solution:

1. **Understand the relationship of probabilities**:
For any two events A and B, the probability of their union can be expressed as:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

2. **Use the independence condition**:
Since A and B are independent, we know that:
\[ P(A \cap B) = P(A) \cdot P(B) \]

3. **Substitute the known values**:
We can substitute \( P(A) \) and \( P(A \cup B) \) into the union formula:
\[ 0.60 = 0.35 + P(B) - P(A) \cdot P(B) \]

4. **Rewrite the equation**:
Substitute \( P(A) \cdot P(B) \) with \( 0.35 \cdot P(B) \):
\[ 0.60 = 0.35 + P(B) - 0.35 \cdot P(B) \]

5. **Combine like terms**:
Rearranging gives us:
\[ 0.60 = 0.35 + P(B)(1 - 0.35) \]
Simplifying \( 1 - 0.35 \) gives us \( 0.65 \):
\[ 0.60 = 0.35 + 0.65P(B) \]

6. **Isolate \( P(B) \)**:
Subtract \( 0.35 \) from both sides:
\[ 0.60 - 0.35 = 0.65P(B) \]
\[ 0.25 = 0.65P(B) \]

7. **Solve for \( P(B) \)**:
Divide both sides by \( 0.65 \):
\[ P(B) = \frac{0.25}{0.65} \]

8. **Simplify the fraction**:
To simplify \( \frac{0.25}{0.65} \):
\[ P(B) = \frac{25}{65} = \frac{5}{13} \]

### Final Answer:
\[ P(B) = \frac{5}{13} \]