(a) Given two independent events A, 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).
(b) If `P(A)=0.2,P(A uu B)=0.6`, find P (B).

Answer

Let's solve the problem step by step.

### Part (a)

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

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

**(ii) Find \( P(A \text{ and not } B) \)**

To find \( P(A \text{ and not } B) \), we can use the formula:
\[ P(A \text{ and not } B) = P(A) - P(A \text{ and } B) \]
Substituting the values:
\[ P(A \text{ and not } B) = 0.3 - 0.18 = 0.12 \]

**(iii) Find \( P(A \text{ or } B) \)**

The probability of A or B is given by:
\[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \]
Substituting the values:
\[ P(A \text{ or } B) = 0.3 + 0.6 - 0.18 = 0.72 \]

**(iv) Find \( P(\text{neither } A \text{ nor } B) \)**

The probability of neither A nor B occurring is:
\[ P(\text{neither } A \text{ nor } B) = 1 - P(A \text{ or } B) \]
Substituting the value from part (iii):
\[ P(\text{neither } A \text{ nor } B) = 1 - 0.72 = 0.28 \]

### Part (b)

Given:
- \( P(A) = 0.2 \)
- \( P(A \cup B) = 0.6 \)

We need to find \( P(B) \).

Using the formula for the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \text{ and } B) \]
Since A and B are independent:
\[ P(A \text{ and } B) = P(A) \times P(B) \]
Substituting the known values:
\[ 0.6 = 0.2 + P(B) - (0.2 \times P(B)) \]
Let \( P(B) = x \):
\[ 0.6 = 0.2 + x - 0.2x \]
Rearranging gives:
\[ 0.6 - 0.2 = x - 0.2x \]
\[ 0.4 = x(1 - 0.2) \]
\[ 0.4 = 0.8x \]
Dividing both sides by 0.8:
\[ x = \frac{0.4}{0.8} = 0.5 \]

### Final Answers:
- (i) \( P(A \text{ and } B) = 0.18 \)
- (ii) \( P(A \text{ and not } B) = 0.12 \)
- (iii) \( P(A \text{ or } B) = 0.72 \)
- (iv) \( P(\text{neither } A \text{ nor } B) = 0.28 \)
- (b) \( P(B) = 0.5 \)