If `P(A)=(6)/(11),P(B)=(5)/(11)` and `P(A uu B)=(7)/(11)`, then
(i) `P(A nn B)`=__________ (ii) `P(B//A)`=_________

Answer

To solve the problem, we need to find two probabilities: \( P(A \cap B) \) and \( P(B|A) \).

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

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

We use the formula for the probability of the union of two events:

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

Substituting the known values into the formula:

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

### Step 2: Simplify the equation

Combine the probabilities on the right side:

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

### Step 3: Rearranging the equation

Now, rearranging the equation to solve for \( P(A \cap B) \):

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

### Step 4: Calculate \( P(A \cap B) \)

Calculating the right side gives:

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

So, the answer for \( P(A \cap B) \) is:

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

### Step 5: Find \( P(B|A) \)

Now we need to find \( P(B|A) \), which is given by the formula:

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

Substituting the values we have:

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

### Step 6: Simplify \( P(B|A) \)

This simplifies to:

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

### Step 7: Final simplification

Reducing \( \frac{4}{6} \):

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

So, the answer for \( P(B|A) \) is:

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

### Final Answers:
(i) \( P(A \cap B) = \frac{4}{11} \)

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

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