If `P(A)=(1)/(4), P(overline(B))=(1)/(2) " and " P(A cup B)=(5)/(9)`, then `P(A//B)` is

Options

A. `(7)/(36)`
B. `(7)/(9)`
C. `(7)/(18)`
D. `(7)/(72)`

Explanation

To solve the problem, we need to find \( P(A|B) \) given the following probabilities:

- \( P(A) = \frac{1}{4} \)
- \( P(\overline{B}) = \frac{1}{2} \)
- \( P(A \cup B) = \frac{5}{9} \)

### Step-by-step Solution:

**Step 1: Find \( P(B) \)**
We know that \( P(\overline{B}) = 1 - P(B) \).
Given \( P(\overline{B}) = \frac{1}{2} \), we can find \( P(B) \):

\[ P(B) = 1 - P(\overline{B}) = 1 - \frac{1}{2} = \frac{1}{2} \]

**Step 2: Use the formula for \( P(A \cup B) \)**
The formula for the union of two events is:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Substituting the known values into the formula:

\[ \frac{5}{9} = P(A) + P(B) - P(A \cap B) \]

Substituting \( P(A) = \frac{1}{4} \) and \( P(B) = \frac{1}{2} \):

\[ \frac{5}{9} = \frac{1}{4} + \frac{1}{2} - P(A \cap B) \]

**Step 3: Convert fractions to a common denominator**
The least common multiple of 4 and 2 is 4. Convert \( \frac{1}{2} \) to have a denominator of 4:

\[ \frac{1}{2} = \frac{2}{4} \]

Now substituting back:

\[ \frac{5}{9} = \frac{1}{4} + \frac{2}{4} - P(A \cap B) \]

This simplifies to:

\[ \frac{5}{9} = \frac{3}{4} - P(A \cap B) \]

**Step 4: Isolate \( P(A \cap B) \)**
Rearranging the equation gives:

\[ P(A \cap B) = \frac{3}{4} - \frac{5}{9} \]

**Step 5: Find a common denominator**
The least common multiple of 4 and 9 is 36. Convert both fractions:

\[ \frac{3}{4} = \frac{27}{36}, \quad \frac{5}{9} = \frac{20}{36} \]

Now substituting these values:

\[ P(A \cap B) = \frac{27}{36} - \frac{20}{36} = \frac{7}{36} \]

**Step 6: Calculate \( P(A|B) \)**
Using the formula for conditional probability:

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

Substituting the known values:

\[ P(A|B) = \frac{\frac{7}{36}}{\frac{1}{2}} = \frac{7}{36} \times \frac{2}{1} = \frac{7 \times 2}{36 \times 1} = \frac{14}{36} = \frac{7}{18} \]

### Final Answer:
\[ P(A|B) = \frac{7}{18} \]