If `P(A cup B)=3//4 " and " P(overline(A))=2//3, " then " P(overline(A) cap B)` is equal to

Options

A. `1//12`
B. `7//12`
C. `5//12`
D. `1//2`

Answer

To solve the problem, we need to find \( P(\overline{A} \cap B) \) given that \( P(A \cup B) = \frac{3}{4} \) and \( P(\overline{A}) = \frac{2}{3} \).

### Step-by-Step Solution:

1. **Find \( P(A) \)**:
We know that \( P(\overline{A}) = 1 - P(A) \).
Given \( P(\overline{A}) = \frac{2}{3} \), we can find \( P(A) \):
\[ P(A) = 1 - P(\overline{A}) = 1 - \frac{2}{3} = \frac{1}{3} \]

**Hint**: Use the relationship between the probability of an event and its complement.

2. **Use the formula for the union of two events**:
The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Rearranging gives us:
\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \]

3. **Substituting known values**:
We know \( P(A \cup B) = \frac{3}{4} \) and \( P(A) = \frac{1}{3} \). We need to express \( P(B) \) in terms of \( P(A \cap B) \):
\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \]
Let \( P(B) = x \):
\[ P(A \cap B) = \frac{1}{3} + x - \frac{3}{4} \]

4. **Finding \( P(A \cap B) \)**:
Rearranging gives:
\[ P(A \cap B) = x - \frac{3}{4} + \frac{1}{3} \]
To combine the fractions, we need a common denominator (which is 12):
\[ P(A \cap B) = x - \frac{9}{12} + \frac{4}{12} = x - \frac{5}{12} \]

5. **Finding \( P(\overline{A} \cap B) \)**:
We can use the relationship:
\[ P(\overline{A} \cap B) = P(B) - P(A \cap B) \]
Substituting \( P(A \cap B) \):
\[ P(\overline{A} \cap B) = x - (x - \frac{5}{12}) = \frac{5}{12} \]

6. **Final Answer**:
Thus, the value of \( P(\overline{A} \cap B) \) is:
\[ \boxed{\frac{5}{12}} \]