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 relationships between events A and B.

### Step 1: Understand the Given Information
We are given:
- \( P(A) = \frac{1}{4} \)
- \( P(A | B) = \frac{1}{4} \)
- \( P(B | A) = \frac{1}{2} \)

### Step 2: Use the Definition of Conditional Probability
From the definition of conditional probability, we know:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
Substituting the known values:
\[ \frac{1}{4} = \frac{P(A \cap B)}{P(B)} \quad \text{(1)} \]

Similarly, for \( P(B | A) \):
\[ P(B | A) = \frac{P(A \cap B)}{P(A)} \]
Substituting the known values:
\[ \frac{1}{2} = \frac{P(A \cap B)}{P(A)} \quad \text{(2)} \]

### Step 3: Solve for \( P(A \cap B) \)
From equation (2), we can express \( P(A \cap B) \):
\[ P(A \cap B) = P(B | A) \cdot P(A) = \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8} \]

### Step 4: Substitute \( P(A \cap B) \) Back into Equation (1)
Now we substitute \( P(A \cap B) \) into equation (1):
\[ \frac{1}{4} = \frac{\frac{1}{8}}{P(B)} \]
Cross-multiplying gives:
\[ P(B) = \frac{1}{8} \cdot 4 = \frac{1}{2} \]

### Step 5: Verify Independence of Events A and B
To check if A and B are independent, we need to see if:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Calculating \( P(A) \cdot P(B) \):
\[ P(A) \cdot P(B) = \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8} \]
Since \( P(A \cap B) = \frac{1}{8} \), we conclude that A and B are independent.

### Step 6: Calculate \( P(A') \) and \( P(B') \)
Now, we can calculate the probabilities of the complements:
\[ P(A') = 1 - P(A) = 1 - \frac{1}{4} = \frac{3}{4} \]
\[ P(B') = 1 - P(B) = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 7: Calculate \( P(A' | B) \) and \( P(B' | A') \)
Using the definition of conditional probability:
\[ P(A' | B) = \frac{P(A' \cap B)}{P(B)} = \frac{P(B) - P(A \cap B)}{P(B)} = \frac{\frac{1}{2} - \frac{1}{8}}{\frac{1}{2}} = \frac{\frac{4}{8} - \frac{1}{8}}{\frac{4}{8}} = \frac{3/8}{4/8} = \frac{3}{4} \]

For \( P(B' | A') \):
\[ P(B' | A') = \frac{P(B')}{P(A')} = \frac{\frac{1}{2}}{\frac{3}{4}} = \frac{2}{3} \]

### Conclusion
Now we can analyze the options provided in the question to determine which statement is not true.