If A and B are independent events such that `P(A)=0.3` and `P(B)=0.5`, then find the values of:
(i) `P(AnnB)` (ii) `P(AuuB)`
(iii) `P(A//B)` (iv) `P(B//A)`

Explanation

To solve the problem step by step, we will find the required probabilities based on the given information that events A and B are independent, with \( P(A) = 0.3 \) and \( P(B) = 0.5 \).

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

For independent events, the probability of the intersection of A and B is given by:

\[ P(A \cap B) = P(A) \times P(B) \]

Substituting the values:

\[ P(A \cap B) = 0.3 \times 0.5 = 0.15 \]

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

The probability of the union of two events A and B is given by:

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

Substituting the values we have:

\[ P(A \cup B) = 0.3 + 0.5 - 0.15 = 0.65 \]

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

The conditional probability of A given B is calculated using the formula:

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

Substituting the values:

\[ P(A | B) = \frac{0.15}{0.5} = 0.3 \]

### Step 4: Find \( P(B | A) \) (Probability of B given A)

Similarly, the conditional probability of B given A is calculated as:

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

Substituting the values:

\[ P(B | A) = \frac{0.15}{0.3} = 0.5 \]

### Final Answers

(i) \( P(A \cap B) = 0.15 \)
(ii) \( P(A \cup B) = 0.65 \)
(iii) \( P(A | B) = 0.3 \)
(iv) \( P(B | A) = 0.5 \)

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