If `P(A)=(1)/(4),P(B) =(1)/(3) "and " P(A nn B)=(1)/(5) " then " P(bar(B) //bar(A))=?`

Options

A. `(11)/(15)`
B. `(11)/(45)`
C. `(23)/(60)`
D. `(37)/(45)`

Explanation

To solve the problem, we need to find \( P(\bar{B} | \bar{A}) \), which is the conditional probability of the complement of event B given the complement of event A.

### Step-by-Step Solution:

1. **Identify Given Probabilities:**
- \( P(A) = \frac{1}{4} \)
- \( P(B) = \frac{1}{3} \)
- \( P(A \cap B) = \frac{1}{5} \)

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 values:
\[ P(A \cup B) = \frac{1}{4} + \frac{1}{3} - \frac{1}{5} \]
To perform this calculation, we need a common denominator. The least common multiple of 4, 3, and 5 is 60.
- Convert each probability:
\[ P(A) = \frac{1}{4} = \frac{15}{60}, \quad P(B) = \frac{1}{3} = \frac{20}{60}, \quad P(A \cap B) = \frac{1}{5} = \frac{12}{60} \]
- Now substitute:
\[ P(A \cup B) = \frac{15}{60} + \frac{20}{60} - \frac{12}{60} = \frac{23}{60} \]

3. **Calculate \( P(\bar{A}) \) and \( P(\bar{B}) \):**
- \( P(\bar{A}) = 1 - P(A) = 1 - \frac{1}{4} = \frac{3}{4} \)
- \( P(\bar{B}) = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3} \)

4. **Calculate \( P(\bar{A} \cap \bar{B}) \):**
Using the formula:
\[ P(\bar{A} \cap \bar{B}) = 1 - P(A \cup B) \]
Substitute the value we found for \( P(A \cup B) \):
\[ P(\bar{A} \cap \bar{B}) = 1 - \frac{23}{60} = \frac{37}{60} \]

5. **Calculate \( P(\bar{B} | \bar{A}) \):**
Using the definition of conditional probability:
\[ P(\bar{B} | \bar{A}) = \frac{P(\bar{B} \cap \bar{A})}{P(\bar{A})} \]
Substitute the values:
\[ P(\bar{B} | \bar{A}) = \frac{P(\bar{A} \cap \bar{B})}{P(\bar{A})} = \frac{\frac{37}{60}}{\frac{3}{4}} \]
To simplify:
\[ P(\bar{B} | \bar{A}) = \frac{37}{60} \times \frac{4}{3} = \frac{148}{180} = \frac{74}{90} = \frac{37}{45} \]

### Final Answer:
\[ P(\bar{B} | \bar{A}) = \frac{37}{45} \]