Given that `P(A) = 1//3, P (B) = 1//4, P (A//B) = 1//6`, then what is `P (B//A) ` equal to?

Options

A. `1//4`
B. `1//8`
C. `3//4`
D. `1//2`

Answer

To find \( P(B|A) \) given \( P(A) = \frac{1}{3} \), \( P(B) = \frac{1}{4} \), and \( P(A|B) = \frac{1}{6} \), we can use the formula for conditional probability:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

From this, we can rearrange the formula to find \( P(A \cap B) \):

\[ P(A \cap B) = P(A|B) \cdot P(B) \]

### Step 1: Calculate \( P(A \cap B) \)

Substituting the known values:

\[ P(A \cap B) = P(A|B) \cdot P(B) = \frac{1}{6} \cdot \frac{1}{4} \]

Calculating this gives:

\[ P(A \cap B) = \frac{1}{24} \]

### Step 2: Use \( P(A \cap B) \) to find \( P(B|A) \)

Now, we can use the formula for \( P(B|A) \):

\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]

Substituting the values we have:

\[ P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{\frac{1}{24}}{\frac{1}{3}} \]

### Step 3: Simplify \( P(B|A) \)

To simplify this, we can multiply by the reciprocal of \( P(A) \):

\[ P(B|A) = \frac{1}{24} \cdot \frac{3}{1} = \frac{3}{24} \]

This simplifies to:

\[ P(B|A) = \frac{1}{8} \]

### Final Answer

Thus, \( P(B|A) = \frac{1}{8} \).