If `P(A) =(6)/(11) ,P(B) =(5)/(11) "and " P(A uu B)=(7)/(11), " then " P(A //B) =?`

Options

A. `(5)/(6)`
B. `(5)/(7)`
C. `(6)/(7)`
D. `(4)/(5)`

Explanation

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

1. **Given Values**:
- \( P(A) = \frac{6}{11} \)
- \( P(B) = \frac{5}{11} \)
- \( P(A \cup B) = \frac{7}{11} \)

2. **Formula for Union**:
The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
We can rearrange this formula to find \( P(A \cap B) \):
\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \]

3. **Substituting the Values**:
Now, we substitute the known values into the rearranged formula:
\[ P(A \cap B) = \frac{6}{11} + \frac{5}{11} - \frac{7}{11} \]

4. **Calculating \( P(A \cap B) \)**:
Simplifying the right side:
\[ P(A \cap B) = \frac{6 + 5 - 7}{11} = \frac{4}{11} \]

5. **Using the Conditional Probability Formula**:
The formula for conditional probability \( P(A | B) \) is:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]

6. **Substituting the Values**:
Now, we substitute \( P(A \cap B) \) and \( P(B) \) into the conditional probability formula:
\[ P(A | B) = \frac{\frac{4}{11}}{\frac{5}{11}} \]

7. **Calculating \( P(A | B) \)**:
Simplifying this expression:
\[ P(A | B) = \frac{4}{11} \times \frac{11}{5} = \frac{4}{5} \]

Thus, the final answer is:
\[ P(A | B) = \frac{4}{5} \]