If `P(A)=0.3, P(B)=0.6` and `P(A//B)=0.4` then find:
(i) `P(AnnB)` (ii) `P(B//A)`

Explanation

To solve the problem, we will use the definitions of conditional probability and the relationship between joint and marginal probabilities.

Given:
- \( P(A) = 0.3 \)
- \( P(B) = 0.6 \)
- \( P(A|B) = 0.4 \)

We need to find:
(i) \( P(A \cap B) \) (the probability of A and B occurring together)
(ii) \( P(B|A) \) (the probability of B occurring given that A has occurred)

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

By the definition of conditional probability, we know:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

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

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

Calculating this:
\[ P(A \cap B) = 0.24 \]

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

Again, using the definition of conditional probability:
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]

We already found \( P(A \cap B) = 0.24 \) and we know \( P(A) = 0.3 \). Substituting these values:
\[ P(B|A) = \frac{0.24}{0.3} \]

Calculating this:
\[ P(B|A) = 0.8 \]

### Final Answers:
(i) \( P(A \cap B) = 0.24 \)
(ii) \( P(B|A) = 0.8 \)

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