If A and B are two events such that `P(A)=1//2 and P(B)=2//3`, then `(A'cap B)` may be

Options

A. `1//2`
B. `1//3`
C. `1//4`
D. `1//5`

Answer

To solve the problem, we need to find the possible values for the probability of the event \( A' \cap B \) given that \( P(A) = \frac{1}{2} \) and \( P(B) = \frac{2}{3} \).

### Step-by-Step Solution:

1. **Identify the given probabilities:**
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{2}{3} \)

2. **Calculate \( P(A') \):**
- The complement of event A, denoted as \( A' \), is given by:
\[ P(A') = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2} \]

3. **Use the formula for the probability of the union of two events:**
- We know that:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- Rearranging gives:
\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \]

4. **Determine the range for \( P(A \cup B) \):**
- The probability of any event must be between 0 and 1:
\[ 0 \leq P(A \cup B) \leq 1 \]

5. **Substitute the known probabilities into the equation:**
- Plugging in the values we have:
\[ P(A \cap B) = \frac{1}{2} + \frac{2}{3} - P(A \cup B) \]
- To combine the fractions, convert \( \frac{1}{2} \) to a fraction with a denominator of 6:
\[ P(A \cap B) = \frac{3}{6} + \frac{4}{6} - P(A \cup B) = \frac{7}{6} - P(A \cup B) \]

6. **Establish the inequalities for \( P(A \cap B) \):**
- Since \( P(A \cup B) \) can range from 0 to 1, we analyze:
- If \( P(A \cup B) = 0 \):
\[ P(A \cap B) = \frac{7}{6} - 0 = \frac{7}{6} \quad \text{(not possible, as probabilities cannot exceed 1)} \]
- If \( P(A \cup B) = 1 \):
\[ P(A \cap B) = \frac{7}{6} - 1 = \frac{1}{6} \]

7. **Final inequalities for \( P(A' \cap B) \):**
- We know that:
\[ P(A' \cap B) = P(B) - P(A \cap B) \]
- Thus:
\[ P(A' \cap B) = \frac{2}{3} - P(A \cap B) \]
- From the earlier steps, we have:
\[ P(A \cap B) \geq \frac{1}{6} \quad \text{and} \quad P(A \cap B) \leq \frac{2}{3} \]
- Therefore:
\[ P(A' \cap B) \geq \frac{2}{3} - \frac{2}{3} = 0 \quad \text{and} \quad P(A' \cap B) \leq \frac{2}{3} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3} \]

8. **Conclusion:**
- The possible range for \( P(A' \cap B) \) is:
\[ 0 \leq P(A' \cap B) \leq \frac{2}{3} \]