For two events `A` and `B`, if `P(A)P((A)/(B))=(1)/(4)` and `P((B)/(A))=(1)/(2)`, then which of the following is not true ?

Options

A. `A` and `B` are independent
B. `P((A')/(B))=(3)/(4)`
C. `P((B')/(A'))=(1)/(2)`
D. none of these

Explanation

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

### Step-by-Step Solution:

1. **Given Information**:
- \( P(A) \cdot P(A|B) = \frac{1}{4} \)
- \( P(B|A) = \frac{1}{2} \)

2. **Expressing \( P(A|B) \)**:
- From the definition of conditional probability:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
- Therefore, we can express \( P(A) \cdot P(A|B) \) as:
\[ P(A) \cdot \frac{P(A \cap B)}{P(B)} = \frac{1}{4} \]

3. **Using \( P(B|A) \)**:
- Again, from the definition of conditional probability:
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{1}{2} \]
- Rearranging gives us:
\[ P(A \cap B) = \frac{1}{2} \cdot P(A) \]

4. **Substituting \( P(A \cap B) \)**:
- Substitute \( P(A \cap B) \) in the equation from step 2:
\[ P(A) \cdot \frac{\frac{1}{2} \cdot P(A)}{P(B)} = \frac{1}{4} \]
- This simplifies to:
\[ \frac{1}{2} \cdot \frac{P(A)^2}{P(B)} = \frac{1}{4} \]
- Rearranging gives:
\[ P(A)^2 = \frac{1}{2} P(B) \]

5. **Finding \( P(B) \)**:
- From \( P(A)^2 = \frac{1}{2} P(B) \), we can express \( P(B) \) in terms of \( P(A) \):
\[ P(B) = 2P(A)^2 \]

6. **Substituting \( P(B) \) back into \( P(A|B) \)**:
- We know \( P(A|B) = \frac{P(A \cap B)}{P(B)} \):
\[ P(A|B) = \frac{\frac{1}{2} P(A)}{2P(A)^2} = \frac{1}{4P(A)} \]
- Setting this equal to the given \( P(A|B) \):
\[ \frac{1}{4P(A)} = P(A|B) \]

7. **Finding \( P(A) \)**:
- From \( P(A) \cdot P(A|B) = \frac{1}{4} \):
\[ P(A) \cdot \frac{1}{4P(A)} = \frac{1}{4} \]
- This confirms \( P(A) \) is consistent.

8. **Independence Check**:
- Events A and B are independent if:
\[ P(A \cap B) = P(A) \cdot P(B) \]
- We already have \( P(A \cap B) = \frac{1}{2} P(A) \) and \( P(B) = 2P(A)^2 \):
\[ \frac{1}{2} P(A) = P(A) \cdot 2P(A)^2 \]
- This confirms independence.

9. **Conclusion**:
- We need to check which of the given statements about A and B is not true. Since we have established that A and B are independent, any statement that contradicts this will be the answer.