Let A and B be two events such that
`P(A)=1/(2),P(B)=pandP(AuuB)=3/(5)`
find 'p' if A and B are independent events.

Answer

To solve the problem, we need to find the value of \( p \) given the probabilities of events \( A \) and \( B \), and the fact that they are independent events.

### Step-by-Step Solution:

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

2. **Using the Formula for Union of Two Events:**
The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

3. **Substituting Known Values:**
We can substitute the known values into the formula:
\[ \frac{3}{5} = \frac{1}{2} + p - P(A \cap B) \]

4. **Finding \( P(A \cap B) \) for Independent Events:**
Since \( A \) and \( B \) are independent, we have:
\[ P(A \cap B) = P(A) \cdot P(B) = \frac{1}{2} \cdot p \]

5. **Substituting \( P(A \cap B) \) into the Equation:**
Now we can substitute \( P(A \cap B) \) back into the union formula:
\[ \frac{3}{5} = \frac{1}{2} + p - \left(\frac{1}{2} \cdot p\right) \]

6. **Simplifying the Equation:**
Rearranging gives:
\[ \frac{3}{5} = \frac{1}{2} + p - \frac{1}{2}p \]
Combine like terms:
\[ \frac{3}{5} = \frac{1}{2} + \frac{1}{2}p \]

7. **Isolating \( p \):**
Subtract \( \frac{1}{2} \) from both sides:
\[ \frac{3}{5} - \frac{1}{2} = \frac{1}{2}p \]

To perform the subtraction, convert \( \frac{1}{2} \) to a fraction with a denominator of 10:
\[ \frac{3}{5} = \frac{6}{10}, \quad \frac{1}{2} = \frac{5}{10} \]
Thus:
\[ \frac{6}{10} - \frac{5}{10} = \frac{1}{10} \]
Therefore:
\[ \frac{1}{10} = \frac{1}{2}p \]

8. **Solving for \( p \):**
Multiply both sides by 2:
\[ p = \frac{1}{10} \cdot 2 = \frac{2}{10} = \frac{1}{5} \]

### Final Answer:
Thus, the value of \( p \) is:
\[ p = \frac{1}{5} \]