A) Define independent event. If A and B are independent event then show that thefollowing pair is also independent:i.A andBii.A andB

A) An independent event in probability theory and statistics is an event that is not affected by other events. In other words, the probability of the event occurring remains the same regardless of whether the other events occur or not.

If A and B are independent events, this means that the occurrence of A does not affect the occurrence of B, and vice versa. This can be mathematically represented as P(A ∩ B) = P(A)P(B), where P(A ∩ B) is the probability of both A and B occurring, P(A) is the probability of A occurring, and P(B) is the probability of B occurring.

i. A and B: As stated above, if A and B are independent, then the occurrence of A does not affect the occurrence of B, and vice versa. This means that A and B are independent.

ii. A and B: This seems to be a repetition of the first point. If it's meant to represent the complement of B (usually denoted as B'), then the statement is also true. If A and B are independent, then A and the complement of B are also independent. This can be represented as P(A ∩ B') = P(A)P(B'), where B' is the complement of B. This is because the occurrence of A does not affect the occurrence of not-B, and vice versa.