If A and B are two events such that `P(A)=0.4, P(B)=0.8` and `P(B//A)=0.6`, then find:
(i) `P(AnnB)` (ii) `P(AuuB)` (iii) `P(A//B)`

Explanation

To solve the problem step by step, we will find the required probabilities using the given information.

### Given:
- \( P(A) = 0.4 \)
- \( P(B) = 0.8 \)
- \( P(B|A) = 0.6 \)

### (i) Find \( P(A \cap B) \)

To find \( P(A \cap B) \), we can use the formula for conditional probability:
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]

Rearranging the formula gives:
\[ P(A \cap B) = P(B|A) \times P(A) \]

Substituting the known values:
\[ P(A \cap B) = 0.6 \times 0.4 = 0.24 \]

### (ii) Find \( P(A \cup B) \)

To find \( P(A \cup B) \), we use the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Substituting the known values:
\[ P(A \cup B) = 0.4 + 0.8 - 0.24 \]
\[ P(A \cup B) = 1.2 - 0.24 = 0.96 \]

### (iii) Find \( P(A|B) \)

To find \( P(A|B) \), we can again use the formula for conditional probability:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

Substituting the known values:
\[ P(A|B) = \frac{0.24}{0.8} \]
\[ P(A|B) = 0.3 \]

### Final Answers:
(i) \( P(A \cap B) = 0.24 \)
(ii) \( P(A \cup B) = 0.96 \)
(iii) \( P(A|B) = 0.3 \)