Final answer:
To prove that events A and B⁻ᶜ, A⁻ᶜ and B, and A⁻ᶜ and B⁻ᶜ are independent, we need to show that P(A AND B⁻ᶜ) = P(A)P(B⁻ᶜ), P(A⁻ᶜ AND B) = P(A⁻ᶜ)P(B), and P(A⁻ᶜ AND B⁻ᶜ) = P(A⁻ᶜ)P(B⁻ᶜ) respectively. By using the complement rule and the independence of A and B, we can prove each statement step by step.
Step-by-step explanation:
To prove that two events are independent, we need to show that P(A AND B) = P(A)P(B). Let's prove the three statements one by one.
(a) To prove that A and B⁻ᶜ are independent, we need to prove that P(A AND B⁻ᶜ) = P(A)P(B⁻ᶜ).
- Using the complement rule: P(B⁻ᶜ) = 1 - P(B)
- From the given information, we know that A and B are independent. So, P(A AND B) = P(A)P(B) (given)
- Using the distributive property: P(A AND B⁻ᶜ) = P(A AND (1 - B)) = P(A)P(1 - B)
- By substituting P(B⁻ᶜ) = 1 - P(B), we get: P(A)P(1 - B) = P(A)P(B⁻ᶜ)
Therefore, we have proved that A and B⁻ᶜ are independent.
(b) To prove that A⁻ᶜ and B are independent, we need to prove that P(A⁻ᶜ AND B) = P(A⁻ᶜ)P(B).
- Using the complement rule: P(A⁻ᶜ) = 1 - P(A)
- From the given information, we know that A and B are independent. So, P(A AND B) = P(A)P(B) (given)
- Using the distributive property: P(A⁻ᶜ AND B) = P((1 - A) AND B) = P(B)P(1 - A)
- By substituting P(A⁻ᶜ) = 1 - P(A), we get: P(B)P(1 - A) = P(A⁻ᶜ)P(B)
Therefore, we have proved that A⁻ᶜ and B are independent.
(c) To prove that A⁻ᶜ and B⁻ᶜ are independent, we need to prove that P(A⁻ᶜ AND B⁻ᶜ) = P(A⁻ᶜ)P(B⁻ᶜ).
- Using the complement rule: P(A⁻ᶜ) = 1 - P(A) and P(B⁻ᶜ) = 1 - P(B)
- From the given information, we know that A and B are independent. So, P(A AND B) = P(A)P(B) (given)
- Using the distributive property: P(A⁻ᶜ AND B⁻ᶜ) = P((1 - A) AND (1 - B)) = P(1 - A)P(1 - B)
- By substituting P(A⁻ᶜ) = 1 - P(A) and P(B⁻ᶜ) = 1 - P(B), we get: P(1 - A)P(1 - B) = P(A⁻ᶜ)P(B⁻ᶜ)
Therefore, we have proved that A⁻ᶜ and B⁻ᶜ are independent.