If A and B are two events such that `P(A) = 1/3, P(B) = 1/4 and P(AcapB) = 1/5, then" "P(barB|barA)=`

Options

A. `37/40`
B. `37/45`
C. `23/40`
D. `23/45`

Answer

To solve the problem, we need to find \( P(\bar{B} | \bar{A}) \) given the probabilities of events A and B and their intersection.

### Step-by-Step Solution:

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

2. **Calculate \( P(A \cup B) \):**
We use the formula for the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the values:
\[ P(A \cup B) = \frac{1}{3} + \frac{1}{4} - \frac{1}{5} \]
To perform this calculation, we need a common denominator. The least common multiple (LCM) of 3, 4, and 5 is 60.

- Convert \( P(A) \):
\[ P(A) = \frac{1}{3} = \frac{20}{60} \]
- Convert \( P(B) \):
\[ P(B) = \frac{1}{4} = \frac{15}{60} \]
- Convert \( P(A \cap B) \):
\[ P(A \cap B) = \frac{1}{5} = \frac{12}{60} \]

Now substituting these into the union formula:
\[ P(A \cup B) = \frac{20}{60} + \frac{15}{60} - \frac{12}{60} = \frac{23}{60} \]

3. **Calculate \( P(A \cup \bar{B}) \):**
We know that:
\[ P(A \cup \bar{B}) = 1 - P(A \cup B) \]
Thus:
\[ P(A \cup \bar{B}) = 1 - \frac{23}{60} = \frac{60 - 23}{60} = \frac{37}{60} \]

4. **Calculate \( P(\bar{A}) \):**
Using the complement rule:
\[ P(\bar{A}) = 1 - P(A) = 1 - \frac{1}{3} = \frac{2}{3} \]

5. **Calculate \( P(\bar{B} | \bar{A}) \):**
We use the conditional probability formula:
\[ P(\bar{B} | \bar{A}) = \frac{P(\bar{A} \cap \bar{B})}{P(\bar{A})} \]
We know that:
\[ P(\bar{A} \cap \bar{B}) = P(\bar{A}) - P(A \cup \bar{B}) = P(\bar{A}) - P(A \cup B) \]
Thus:
\[ P(\bar{B} | \bar{A}) = \frac{P(A \cup \bar{B})}{P(\bar{A})} = \frac{\frac{37}{60}}{\frac{2}{3}} = \frac{37}{60} \times \frac{3}{2} = \frac{37 \times 3}{60 \times 2} = \frac{111}{120} = \frac{37}{40} \]

### Final Answer:
\[ P(\bar{B} | \bar{A}) = \frac{37}{40} \]