Final answer:
In probability theory, a sure event and an impossible event are independent of any other event. This is because for a sure event which has a probability of 1, its intersection with any event B is just B, and for an impossible event which has a probability of 0, its intersection with any event B is null.
Step-by-step explanation:
In probability theory, two events are said to be independent if the probability of one event occurring does not alter the probability of the other event occurring. Usually, the mathematical way to verify independence is through the formula P(A ∩ B) = P(A)P(B). Here, if A and B are independent events, the probability of both occurring (P(A ∩ B)) is equal to the product of their individual probabilities (P(A) and P(B)).
In the cases of a sure event and an impossible event, things are a bit different. A sure event is an event that will definitely occur (P(A) = 1), and an impossible event is an event that cannot occur (P(A) = 0).
For any other event B, the intersection of B with a sure event A is just B (since A includes all possible outcomes), so P(A ∩ B) = P(B) = P(A)P(B) because P(A) = 1. Hence, a sure event is independent of any other event. Similarly, for an impossible event A, P(A ∩ B) = P(Ø) = 0 = P(A)P(B) since P(A) = 0. Thus, an impossible event is also independent of any other event.
Learn more about Event Independence