For two events A and B `P(A)=6/13, P(B)=5/13` and `P(AuuB)=7/13`. Find the values of following:
(i) `P(AnnB)` (ii) `P(A//B)`
(iii) `P(B//A)`

Explanation

To solve the problem, we will follow these steps:

### Given:
- \( P(A) = \frac{6}{13} \)
- \( P(B) = \frac{5}{13} \)
- \( P(A \cup B) = \frac{7}{13} \)

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

Using 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:
\[ \frac{7}{13} = \frac{6}{13} + \frac{5}{13} - P(A \cap B) \]

Now, simplify the equation:
\[ \frac{7}{13} = \frac{11}{13} - P(A \cap B) \]

Rearranging gives:
\[ P(A \cap B) = \frac{11}{13} - \frac{7}{13} = \frac{4}{13} \]

### Step 2: Find \( P(A | B) \)

The conditional probability \( P(A | B) \) is given by:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]

Substituting the values we found:
\[ P(A | B) = \frac{\frac{4}{13}}{\frac{5}{13}} = \frac{4}{5} \]

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

The conditional probability \( P(B | A) \) is given by:
\[ P(B | A) = \frac{P(A \cap B)}{P(A)} \]

Substituting the values:
\[ P(B | A) = \frac{\frac{4}{13}}{\frac{6}{13}} = \frac{4}{6} = \frac{2}{3} \]

### Final Results:
(i) \( P(A \cap B) = \frac{4}{13} \)
(ii) \( P(A | B) = \frac{4}{5} \)
(iii) \( P(B | A) = \frac{2}{3} \)

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