Given two independent events A and B such that`P(A) = 0. 3`,`P(B) = 0. 6`. Find(i) P(A and B) (ii) P(A and not B)(iii) P(A or B) (iv) P(neither A nor B)

Explanation

To solve the problem step by step, we will use the properties of independent events and the basic rules of probability.

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

### Step 1: Find \( P(A \text{ and } B) \)

Since A and B are independent events, the probability of both A and B occurring is given by:

\[ P(A \text{ and } B) = P(A) \times P(B) \]

Substituting the values:

\[ P(A \text{ and } B) = 0.3 \times 0.6 = 0.18 \]

### Step 2: Find \( P(A \text{ and not } B) \)

To find the probability of A occurring and B not occurring, we need to calculate \( P(B') \) (the complement of B):

\[ P(B') = 1 - P(B) = 1 - 0.6 = 0.4 \]

Now, we can find \( P(A \text{ and not } B) \):

\[ P(A \text{ and not } B) = P(A) \times P(B') = 0.3 \times 0.4 = 0.12 \]

### Step 3: Find \( P(A \text{ or } B) \)

The probability of A or B occurring is given by the formula:

\[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \]

Substituting the values we have:

\[ P(A \text{ or } B) = 0.3 + 0.6 - 0.18 = 0.72 \]

### Step 4: Find \( P(\text{neither } A \text{ nor } B) \)

The probability of neither A nor B occurring is the complement of the probability of A or B:

\[ P(\text{neither } A \text{ nor } B) = 1 - P(A \text{ or } B) \]

Substituting the value we found:

\[ P(\text{neither } A \text{ nor } B) = 1 - 0.72 = 0.28 \]

### Final Answers:
1. \( P(A \text{ and } B) = 0.18 \)
2. \( P(A \text{ and not } B) = 0.12 \)
3. \( P(A \text{ or } B) = 0.72 \)
4. \( P(\text{neither } A \text{ nor } B) = 0.28 \)

---