If `P(A) = (2)/(5), P(B) = (1)/(3) , P(A cap B) = (1)/(5)`, Find `P((bar(A))/(bar(B)))`.

Answer

To solve the problem, we need to find \( P(\bar{A} | \bar{B}) \) using the given probabilities:

1. **Given Values**:
- \( P(A) = \frac{2}{5} \)
- \( P(B) = \frac{1}{3} \)
- \( P(A \cap B) = \frac{1}{5} \)

2. **Finding \( P(A \cup B) \)**:
We can use the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the values:
\[ P(A \cup B) = \frac{2}{5} + \frac{1}{3} - \frac{1}{5} \]

To add these fractions, we need a common denominator. The least common multiple of 5 and 3 is 15.

Rewriting the fractions:
\[ P(A) = \frac{2}{5} = \frac{6}{15}, \quad P(B) = \frac{1}{3} = \frac{5}{15}, \quad P(A \cap B) = \frac{1}{5} = \frac{3}{15} \]

Now substituting back:
\[ P(A \cup B) = \frac{6}{15} + \frac{5}{15} - \frac{3}{15} = \frac{8}{15} \]

3. **Finding \( P(\bar{A} \cap \bar{B}) \)**:
We can use the formula:
\[ P(\bar{A} \cap \bar{B}) = 1 - P(A \cup B) \]
Substituting the value we found:
\[ P(\bar{A} \cap \bar{B}) = 1 - \frac{8}{15} = \frac{7}{15} \]

4. **Finding \( P(\bar{B}) \)**:
We can use the formula:
\[ P(\bar{B}) = 1 - P(B) \]
Substituting the value:
\[ P(\bar{B}) = 1 - \frac{1}{3} = \frac{2}{3} \]

5. **Finding \( P(\bar{A} | \bar{B}) \)**:
We can use the formula:
\[ P(\bar{A} | \bar{B}) = \frac{P(\bar{A} \cap \bar{B})}{P(\bar{B})} \]
Substituting the values we found:
\[ P(\bar{A} | \bar{B}) = \frac{\frac{7}{15}}{\frac{2}{3}} \]

To divide the fractions, we multiply by the reciprocal:
\[ P(\bar{A} | \bar{B}) = \frac{7}{15} \times \frac{3}{2} = \frac{7 \times 3}{15 \times 2} = \frac{21}{30} = \frac{7}{10} \]

Thus, the final answer is:
\[ P(\bar{A} | \bar{B}) = \frac{7}{10} \]