If `P(A) = 2/3, P(B) = 4/9` and `P(AcapB)=cap`, then `P(A'capB')` is greater than or equal to`

Options

A. `37/45`
B. `1/5`
C. `4/5`
D. `41/45`

Answer

To solve the problem, we need to find \( P(A' \cap B') \) given \( P(A) = \frac{2}{3} \), \( P(B) = \frac{4}{9} \), and \( P(A \cap B) = \frac{14}{45} \).

### Step-by-Step Solution:

1. **Use De Morgan's Law**:
According to De Morgan's Law, we have:
\[ P(A' \cap B') = P((A \cup B)') \]
This means that the probability of the complement of the union of A and B is equal to the intersection of the complements of A and B.

2. **Calculate \( P(A \cup B) \)**:
The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the given values:
\[ P(A \cup B) = \frac{2}{3} + \frac{4}{9} - \frac{14}{45} \]

3. **Finding a common denominator**:
The common denominator for \( \frac{2}{3} \), \( \frac{4}{9} \), and \( \frac{14}{45} \) is 45. We convert each fraction:
- \( \frac{2}{3} = \frac{30}{45} \)
- \( \frac{4}{9} = \frac{20}{45} \)
- \( \frac{14}{45} = \frac{14}{45} \)

4. **Substituting back into the union formula**:
Now we can substitute these values into the union formula:
\[ P(A \cup B) = \frac{30}{45} + \frac{20}{45} - \frac{14}{45} = \frac{30 + 20 - 14}{45} = \frac{36}{45} \]

5. **Calculate \( P(A' \cap B') \)**:
Now, we can find \( P(A' \cap B') \):
\[ P(A' \cap B') = 1 - P(A \cup B) = 1 - \frac{36}{45} = \frac{9}{45} = \frac{1}{5} \]

6. **Conclusion**:
Therefore, \( P(A' \cap B') = \frac{1}{5} \).

### Final Answer:
The probability \( P(A' \cap B') \) is equal to \( \frac{1}{5} \).