Let A and B be two events such that P(A|B) =`1/2`, P(B|A)=`1/3` and `P(A nn B)=1/6`, then which one of the following is not true?

Options

A. `P(A uuB)=2/3`
B. A and B are independent
C. A and B are not independent
D. `P(A' nn B)=1/6`

Answer

To solve the problem, we need to analyze the given probabilities and determine which statement about the events A and B is not true.

### Given Information:
1. \( P(A|B) = \frac{1}{2} \)
2. \( P(B|A) = \frac{1}{3} \)
3. \( P(A \cap B) = \frac{1}{6} \)

### Step 1: Use the definition of conditional probability

The conditional probability \( P(A|B) \) can be expressed as:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Substituting the known values:
\[ \frac{1}{2} = \frac{\frac{1}{6}}{P(B)} \]

### Step 2: Solve for \( P(B) \)

Cross-multiplying gives:
\[ P(B) = \frac{1}{6} \times 2 = \frac{1}{3} \]

### Step 3: Use the definition of conditional probability for \( P(B|A) \)

Similarly, for \( P(B|A) \):
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]
Substituting the known values:
\[ \frac{1}{3} = \frac{\frac{1}{6}}{P(A)} \]

### Step 4: Solve for \( P(A) \)

Cross-multiplying gives:
\[ P(A) = \frac{1}{6} \times 3 = \frac{1}{2} \]

### Step 5: Verify independence of A and B

To check if A and B are independent, we need to verify if:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Substituting the values we found:
\[ P(A) = \frac{1}{2}, \quad P(B) = \frac{1}{3} \]
Calculating \( P(A) \cdot P(B) \):
\[ P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6} \]

### Step 6: Conclusion

Since \( P(A \cap B) = \frac{1}{6} \) and \( P(A) \cdot P(B) = \frac{1}{6} \), we conclude that A and B are independent events.

### Final Statement

Now, we need to identify which statement is not true. The statement that A and B are independent is true. Therefore, any statement claiming they are not independent would be false.