If P(A) = 3/7, P(B) = 1/2 and `P(A'capB')=1//14`, then are A and B are mutually exclusive events?

Options

A. No
B. Yes
C. Either yes or no
D. Cannot be determined

Answer

To determine if events A and B are mutually exclusive, we need to analyze the given probabilities and apply the definitions and properties of probability.

### Step-by-Step Solution:

1. **Understand Definitions**:
- Mutually exclusive events are events that cannot occur at the same time. This means that the probability of their intersection is zero: \( P(A \cap B) = 0 \).

2. **Given Probabilities**:
- \( P(A) = \frac{3}{7} \)
- \( P(B) = \frac{1}{2} \)
- \( P(A' \cap B') = \frac{1}{14} \) (where \( A' \) and \( B' \) are the complements of A and B respectively)

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

4. **Calculate \( P(A \cup B) \)**:
- Since \( P(A' \cap B') = \frac{1}{14} \), we can find \( P(A \cup B) \) using the relationship:
\[ P(A \cup B) + P(A' \cap B') = 1 \]
- Substituting the known value:
\[ P(A \cup B) + \frac{1}{14} = 1 \]
- Therefore:
\[ P(A \cup B) = 1 - \frac{1}{14} = \frac{14}{14} - \frac{1}{14} = \frac{13}{14} \]

5. **Use the Formula for Union**:
- The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- We can substitute the known values:
\[ \frac{13}{14} = \frac{3}{7} + \frac{1}{2} - P(A \cap B) \]

6. **Convert to a Common Denominator**:
- Convert \( P(A) \) and \( P(B) \) to a common denominator (14):
\[ P(A) = \frac{3}{7} = \frac{6}{14}, \quad P(B) = \frac{1}{2} = \frac{7}{14} \]
- Now substitute:
\[ \frac{13}{14} = \frac{6}{14} + \frac{7}{14} - P(A \cap B) \]
- Simplifying gives:
\[ \frac{13}{14} = \frac{13}{14} - P(A \cap B) \]

7. **Solve for \( P(A \cap B) \)**:
- Rearranging gives:
\[ P(A \cap B) = 0 \]

8. **Conclusion**:
- Since \( P(A \cap B) = 0 \), we conclude that events A and B are mutually exclusive.

### Final Answer:
Yes, events A and B are mutually exclusive.