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 use the given probabilities and the formulas for conditional probability, intersection, and union of events.

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

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

Using the formula for conditional probability:
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]
We can rearrange this to find \( P(A \cap B) \):
\[ P(A \cap B) = P(B|A) \times P(A) \]
Substituting the known values:
\[ P(A \cap B) = 0.6 \times 0.4 = 0.24 \]

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

Using the formula for the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the known values:
\[ P(A \cup B) = 0.4 + 0.8 - 0.24 = 1.2 - 0.24 = 0.96 \]

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

Using the formula for conditional probability again:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Substituting the known values:
\[ P(A|B) = \frac{0.24}{0.8} = 0.3 \]

### Summary of Results:
1. \( P(A \cap B) = 0.24 \)
2. \( P(A \cup B) = 0.96 \)
3. \( P(A|B) = 0.3 \)