Question

Consider the following statements. Statement (I): If \( E \) and \( F \) are two independent events, then \( E' \) and \( F' \) are also independent. Statement (II): Two mutually exclusive events with non-zero probabilities of occurrence cannot be independent. Which of the following is correct?

• A. Statement (I) is false and statement (II) is true.
• B. Both the statements are true.
• C. Both the statements are false.
• D. Statement (I) is true and statement (II) is false.

Hint:

When analyzing independent events and their complements, remember that independence for complements follows from the basic properties of probability. Also, mutually exclusive events with non-zero probabilities are never independent.

Answer 1

The Correct Option is B

- Statement (I) is true. If two events \( E \) and \( F \) are independent, then their complements \( E' \) and \( F' \) are also independent. This follows from the fact that the probability of the intersection of \( E' \) and \( F' \) is equal to the product of their individual probabilities, i.e., \( P(E' \cap F') = P(E')P(F') \). - Statement (II) is also true. If two events are mutually exclusive (meaning their intersection is empty), their probabilities cannot be independent unless at least one of them has zero probability. If the probability of both events is non-zero, then they are not independent, because the occurrence of one affects the probability of the other. Thus, both statements (I) and (II) are true.