If P (A) = P (B) = x and `P(AnnB)=P(A'nnB')=1//3`, then x = ?

Options

A. `1//2`
B. `1//3`
C. `1//4`
D. `1//6`

Answer

To solve the problem, we need to find the value of \( x \) given that \( P(A) = P(B) = x \) and \( P(A \cap B) = P(A' \cap B') = \frac{1}{3} \).

### Step-by-Step Solution:

1. **Understanding the Events**:
- We know that \( P(A) = x \) and \( P(B) = x \).
- The probability of the intersection of events \( A \) and \( B \) is given as \( P(A \cap B) \).
- The probability of neither \( A \) nor \( B \) occurring is given as \( P(A' \cap B') = \frac{1}{3} \).

2. **Using the Complement Rule**:
- We can use the formula for the total probability:
\[ P(A \cup B) + P(A' \cap B') = 1 \]
- Thus, we have:
\[ P(A \cup B) = 1 - P(A' \cap B') = 1 - \frac{1}{3} = \frac{2}{3} \]

3. **Using the Inclusion-Exclusion Principle**:
- The probability of the union of two events can be expressed as:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- Substituting the known values:
\[ P(A \cup B) = x + x - P(A \cap B) = 2x - P(A \cap B) \]

4. **Substituting Known Values**:
- We know from the problem that \( P(A \cap B) = \frac{1}{3} \).
- Therefore, we can substitute this into our equation:
\[ \frac{2}{3} = 2x - \frac{1}{3} \]

5. **Solving for \( x \)**:
- Rearranging the equation gives:
\[ 2x = \frac{2}{3} + \frac{1}{3} = \frac{3}{3} = 1 \]
- Dividing both sides by 2:
\[ x = \frac{1}{2} \]

### Final Answer:
Thus, the value of \( x \) is \( \frac{1}{2} \).

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