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Yes, the impossible event (the empty set) is independent of any event. Because its probability is 0, the probability of its intersection with any event is 0, which perfectly satisfies the mathematical definition of independence.

Conditions: Event B is the impossible event ($B = \emptyset$).; Event A is any arbitrary event.

Yes, the impossible event (the empty set) is independent of any event. Because its probability is 0, the probability of its intersection with any event is 0, which perfectly satisfies the mathematical definition of independence.

Conditions: Event B is the impossible event ($B = \emptyset$).; Event A is any arbitrary event.

By defining heads and tails as two events in a single coin toss, they are mutually exclusive because they cannot happen simultaneously. Knowing that one outcome occurred changes the probability of the other to zero, proving they are dependent, not independent.

Conditions: A single toss of a fair coin.; Event A is 'heads up' and Event B is 'tails up'.

By defining heads and tails as two events in a single coin toss, they are mutually exclusive because they cannot happen simultaneously. Knowing that one outcome occurred changes the probability of the other to zero, proving they are dependent, not independent.

Conditions: A single toss of a fair coin.; Event A is 'heads up' and Event B is 'tails up'.

The correct statement is that if the intersection of two events is nonempty ($AB \neq \emptyset$), then the two events may be independent.

Conditions: A and B are any two events in a random experiment.

The correct statement is that if the intersection of two events is nonempty ($AB \neq \emptyset$), then the two events may be independent.

Conditions: A and B are any two events in a random experiment.

Mutually exclusive events cannot occur simultaneously. If event A occurs, event B is guaranteed not to occur, meaning the occurrence of A directly affects the probability of B.

Conditions: Events A and B are mutually exclusive ($AB = \emptyset$).; The probability of event A is greater than 0 ($P(A) > 0$).; The probability of event B is greater than 0 ($P(B) > 0$).

Mutually exclusive events cannot occur simultaneously. If event A occurs, event B is guaranteed not to occur, meaning the occurrence of A directly affects the probability of B.

Conditions: Events A and B are mutually exclusive ($AB = \emptyset$).; The probability of event A is greater than 0 ($P(A) > 0$).; The probability of event B is greater than 0 ($P(B) > 0$).

Two events are mutually exclusive if their intersection is the empty set, meaning they cannot occur simultaneously in the same random experiment.

Conditions: Events A and B are two events in the same random experiment.

Two events are mutually exclusive if their intersection is the empty set, meaning they cannot occur simultaneously in the same random experiment.

Conditions: Events A and B are two events in the same random experiment.