If A and B are two events such that `P(A)=6/11, P(B)=5/11 and P(A cup B)=7/11` find `P(B/A)`. a) 2/3 b) 1/3 c) 1 d) none of these

Options

A. 2/3
B. 1/3
C. 1
D. none of these

Answer

To find \( P(B|A) \), we can use the formula:

\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]

### Step 1: Find \( P(A \cap B) \)

We can calculate \( P(A \cap B) \) using the formula:

\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \]

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

Substituting the values into the formula:

\[ P(A \cap B) = \frac{6}{11} + \frac{5}{11} - \frac{7}{11} \]

### Step 2: Simplify the expression

Combine the fractions:

\[ P(A \cap B) = \frac{6 + 5 - 7}{11} = \frac{4}{11} \]

### Step 3: Calculate \( P(B|A) \)

Now that we have \( P(A \cap B) \), we can substitute it back into the formula for \( P(B|A) \):

\[ P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{\frac{4}{11}}{\frac{6}{11}} \]

### Step 4: Simplify \( P(B|A) \)

When dividing fractions, we multiply by the reciprocal:

\[ P(B|A) = \frac{4}{11} \times \frac{11}{6} = \frac{4}{6} \]

### Step 5: Reduce the fraction

Now, simplify \( \frac{4}{6} \):

\[ P(B|A) = \frac{2}{3} \]

### Final Answer

Thus, the probability \( P(B|A) \) is:

\[ \boxed{\frac{2}{3}} \]