If `P(A)=(2)/(5),P(B)=(3)/(10)` and `P(AcapB)=(1)/(5)` then `P((A')/(B')).P((B')/(A'))` equals

Options

A. `(5)/(6)`
B. `(5)/(7)`
C. `(25)/(42)`
D. `(1)/(42)`

Answer

To solve the problem, we need to find \( P(A' \cap B') \) and then compute \( P(A'|B') \cdot P(B'|A') \).

### Step-by-Step Solution:

1. **Identify Given Values**:
- \( P(A) = \frac{2}{5} \)
- \( P(B) = \frac{3}{10} \)
- \( P(A \cap B) = \frac{1}{5} \)

2. **Calculate \( P(A \cup B) \)**:
Using the formula for the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substitute the given values:
\[ P(A \cup B) = \frac{2}{5} + \frac{3}{10} - \frac{1}{5} \]
To perform the addition, convert all fractions to have a common denominator (10):
\[ P(A \cup B) = \frac{4}{10} + \frac{3}{10} - \frac{2}{10} = \frac{5}{10} = \frac{1}{2} \]

3. **Calculate \( P(A' \cap B') \)**:
Using De Morgan's law:
\[ P(A' \cap B') = 1 - P(A \cup B) \]
Substitute the value we found for \( P(A \cup B) \):
\[ P(A' \cap B') = 1 - \frac{1}{2} = \frac{1}{2} \]

4. **Calculate \( P(B') \)**:
Using the complement rule:
\[ P(B') = 1 - P(B) = 1 - \frac{3}{10} = \frac{7}{10} \]

5. **Calculate \( P(A') \)**:
Using the complement rule:
\[ P(A') = 1 - P(A) = 1 - \frac{2}{5} = \frac{3}{5} \]

6. **Calculate \( P(A' | B') \)**:
Using the conditional probability formula:
\[ P(A' | B') = \frac{P(A' \cap B')}{P(B')} \]
Substitute the values:
\[ P(A' | B') = \frac{\frac{1}{2}}{\frac{7}{10}} = \frac{1}{2} \cdot \frac{10}{7} = \frac{5}{7} \]

7. **Calculate \( P(B' | A') \)**:
Using the conditional probability formula:
\[ P(B' | A') = \frac{P(A' \cap B')}{P(A')} \]
Substitute the values:
\[ P(B' | A') = \frac{\frac{1}{2}}{\frac{3}{5}} = \frac{1}{2} \cdot \frac{5}{3} = \frac{5}{6} \]

8. **Final Calculation**:
Now, we need to find \( P(A' | B') \cdot P(B' | A') \):
\[ P(A' | B') \cdot P(B' | A') = \frac{5}{7} \cdot \frac{5}{6} = \frac{25}{42} \]

### Final Answer:
\[ P(A' | B') \cdot P(B' | A') = \frac{25}{42} \]