If A and B are two independent events such that `P(A)=(1)/(2),P(A uu B)=(3)/(5)` and `P(B)=p`, then p = _________.

Answer

To solve the problem, we need to find the probability \( P(B) = p \) given the probabilities of events A and B and their union.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- \( P(A) = \frac{1}{2} \)
- \( P(A \cup B) = \frac{3}{5} \)
- \( P(B) = p \)

2. **Use the Formula for the Union of Two Events:**
The formula for the probability of the union of two independent events A and B is given by:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Since A and B are independent, we have:
\[ P(A \cap B) = P(A) \cdot P(B) = P(A) \cdot p \]

3. **Substitute the Known Values into the Formula:**
Substitute \( P(A) \) and \( P(A \cup B) \) into the union formula:
\[ \frac{3}{5} = \frac{1}{2} + p - \left(\frac{1}{2} \cdot p\right) \]

4. **Simplify the Equation:**
Rearranging the equation gives:
\[ \frac{3}{5} = \frac{1}{2} + p - \frac{1}{2}p \]
Combine like terms:
\[ \frac{3}{5} = \frac{1}{2} + \left(1 - \frac{1}{2}\right)p \]
This simplifies to:
\[ \frac{3}{5} = \frac{1}{2} + \frac{1}{2}p \]

5. **Isolate \( p \):**
To isolate \( p \), first subtract \( \frac{1}{2} \) from both sides:
\[ \frac{3}{5} - \frac{1}{2} = \frac{1}{2}p \]
To perform the subtraction, find a common denominator (which is 10):
\[ \frac{3}{5} = \frac{6}{10}, \quad \frac{1}{2} = \frac{5}{10} \]
Thus:
\[ \frac{6}{10} - \frac{5}{10} = \frac{1}{10} \]
So we have:
\[ \frac{1}{10} = \frac{1}{2}p \]

6. **Solve for \( p \):**
Multiply both sides by 2 to solve for \( p \):
\[ p = 2 \cdot \frac{1}{10} = \frac{2}{10} = \frac{1}{5} \]

### Final Answer:
\[ p = \frac{1}{5} \]