Consider the following statements :
if A and B are independent events, then
1. A and `bar(B)` are independent.
2. `barA` and B are independent.
3. `barA` and `barB` are independent.
Which of the above statements is/are correct ?

Options

A. 3 only
B. 1 and 2 only
C. 1, 2 and 3
D. None of these

Explanation

To determine which statements are correct regarding the independence of events A and B, we will analyze each statement one by one.

### Given Information:
- A and B are independent events.

### Definitions:
- Two events A and B are independent if:
\[ P(A \cap B) = P(A) \cdot P(B) \]
- The complement of an event A is denoted as \( \bar{A} \) and represents the event that A does not occur.

### Statement 1: A and \( \bar{B} \) are independent.
To prove this, we need to show:
\[ P(A \cap \bar{B}) = P(A) \cdot P(\bar{B}) \]

Using the property of complements:
\[ P(\bar{B}) = 1 - P(B) \]
Now, we can express \( P(A \cap \bar{B}) \) as:
\[ P(A \cap \bar{B}) = P(A) - P(A \cap B) \]
Since A and B are independent, we can substitute:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Thus,
\[ P(A \cap \bar{B}) = P(A) - P(A) \cdot P(B) = P(A)(1 - P(B)) = P(A) \cdot P(\bar{B}) \]
This shows that A and \( \bar{B} \) are independent.

### Statement 2: \( \bar{A} \) and B are independent.
To prove this, we need to show:
\[ P(\bar{A} \cap B) = P(\bar{A}) \cdot P(B) \]

Using the property of complements:
\[ P(\bar{A}) = 1 - P(A) \]
Now, we can express \( P(\bar{A} \cap B) \) as:
\[ P(\bar{A} \cap B) = P(B) - P(A \cap B) \]
Again, using the independence of A and B, we substitute:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Thus,
\[ P(\bar{A} \cap B) = P(B) - P(A) \cdot P(B) = P(B)(1 - P(A)) = P(B) \cdot P(\bar{A}) \]
This shows that \( \bar{A} \) and B are independent.

### Statement 3: \( \bar{A} \) and \( \bar{B} \) are independent.
To prove this, we need to show:
\[ P(\bar{A} \cap \bar{B}) = P(\bar{A}) \cdot P(\bar{B}) \]

Using the property of complements:
\[ P(\bar{A} \cap \bar{B}) = 1 - P(A \cup B) \]
Using the formula for the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) = P(A) + P(B) - P(A) \cdot P(B) \]
Thus,
\[ P(\bar{A} \cap \bar{B}) = 1 - (P(A) + P(B) - P(A) \cdot P(B)) = 1 - P(A) - P(B) + P(A) \cdot P(B) \]
Now, substituting \( P(\bar{A}) \) and \( P(\bar{B}) \):
\[ P(\bar{A}) = 1 - P(A) \quad \text{and} \quad P(\bar{B}) = 1 - P(B) \]
Thus,
\[ P(\bar{A}) \cdot P(\bar{B}) = (1 - P(A))(1 - P(B)) = 1 - P(A) - P(B) + P(A) \cdot P(B) \]
This shows that \( \bar{A} \) and \( \bar{B} \) are independent.

### Conclusion:
All three statements are correct. Therefore, the answer is:
1. Statement 1 is correct.
2. Statement 2 is correct.
3. Statement 3 is correct.

### Final Answer:
All statements (1, 2, and 3) are correct.

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