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 start with the given information:

1. \( P(A) = P(B) = x \)
2. \( P(A \cap B) = \frac{1}{3} \)
3. \( P(A' \cap B') = \frac{1}{3} \)

We need to find the value of \( x \).

### Step 1: Use the formula for the probability of the union of two events.

We know that:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Substituting the known values:
\[ P(A \cup B) = x + x - \frac{1}{3} = 2x - \frac{1}{3} \]

### Step 2: Use the relationship between the complement and the union.

We also know that:
\[ P(A' \cap B') = 1 - P(A \cup B) \]

Substituting the value of \( P(A' \cap B') \):
\[ \frac{1}{3} = 1 - P(A \cup B) \]

### Step 3: Substitute \( P(A \cup B) \) from Step 1 into the equation.

From Step 2, we can express \( P(A \cup B) \):
\[ P(A \cup B) = 1 - \frac{1}{3} = \frac{2}{3} \]

Now, we can set the two expressions for \( P(A \cup B) \) equal to each other:
\[ 2x - \frac{1}{3} = \frac{2}{3} \]

### Step 4: Solve for \( x \).

To solve for \( x \), we first add \( \frac{1}{3} \) to both sides:
\[ 2x = \frac{2}{3} + \frac{1}{3} \]
\[ 2x = \frac{3}{3} = 1 \]

Now, divide both sides by 2:
\[ x = \frac{1}{2} \]

### Conclusion

Thus, the value of \( x \) is:
\[ \boxed{\frac{1}{2}} \]