If A and B are two independent events such that `P(bar(A))`= 0.65, `P( A uu B)`= 0.65 and P(B)= p, the value of p is

Options

A. `(1)/(13)`
B. `(3)/(13)`
C. `(6)/(13)`
D. None of these

Answer

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

### Step-by-Step Solution:

1. **Identify Given Information:**
- \( P(\bar{A}) = 0.65 \)
- \( P(A \cup B) = 0.65 \)
- \( P(B) = p \)

2. **Calculate \( P(A) \):**
- We know that \( P(\bar{A}) = 1 - P(A) \).
- Therefore, \( P(A) = 1 - P(\bar{A}) = 1 - 0.65 = 0.35 \).

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

4. **Substitute Known Values into the Union Formula:**
- Now substituting the known values into the union formula:
\[ 0.65 = P(A) + P(B) - P(A \cap B) \]
- This becomes:
\[ 0.65 = 0.35 + p - (0.35 \cdot p) \]

5. **Simplify the Equation:**
- Rearranging gives:
\[ 0.65 = 0.35 + p - 0.35p \]
- Combine like terms:
\[ 0.65 = 0.35 + p(1 - 0.35) \]
- This simplifies to:
\[ 0.65 = 0.35 + 0.65p \]

6. **Isolate \( p \):**
- Subtract \( 0.35 \) from both sides:
\[ 0.65 - 0.35 = 0.65p \]
- This gives:
\[ 0.30 = 0.65p \]

7. **Solve for \( p \):**
- Divide both sides by \( 0.65 \):
\[ p = \frac{0.30}{0.65} \]
- To simplify, multiply numerator and denominator by 100:
\[ p = \frac{30}{65} \]
- Simplifying \( \frac{30}{65} \) by dividing both by 5:
\[ p = \frac{6}{13} \]

### Final Answer:
Thus, the value of \( p \) is \( \frac{6}{13} \).