How is independence defined in terms of probability?

How is independence defined in terms of probability?

• A. The occurrence of an event does not impact another event
• B. Events always happen simultaneously
• C. The sum of probabilities equals 1
• D. At least one event always occurs

Answer: (A)

Independence in probability is defined such that the occurrence of one event does not affect the occurrence of another event. This means that if two events are independent, the probability of both events occurring together is simply the product of their individual probabilities. Formally, if events A and B are independent, then \( P(A \cap B) = P(A) \times P(B) \). This definition captures the essence of independence because it illustrates that knowing the outcome of one event provides no information about the outcome of the other. For instance, if you were to flip a coin and roll a die simultaneously, the result of the coin flip does not influence the result of the die roll. The other options do not accurately describe independence. The idea of events always happening simultaneously is unrelated to independence; independence can exist even if the events occur at different times. The statement about the sum of probabilities equaling 1 relates to the axioms of probability rather than to independence itself. The notion of at least one event always occurring pertains to certain types of events but does not define independent events. Thus, the definition of independence focuses specifically on the lack of influence between events.

Independence in probability is defined such that the occurrence of one event does not affect the occurrence of another event. This means that if two events are independent, the probability of both events occurring together is simply the product of their individual probabilities. Formally, if events A and B are independent, then ( P(A \cap B) = P(A) \times P(B) ).

This definition captures the essence of independence because it illustrates that knowing the outcome of one event provides no information about the outcome of the other. For instance, if you were to flip a coin and roll a die simultaneously, the result of the coin flip does not influence the result of the die roll.

The other options do not accurately describe independence. The idea of events always happening simultaneously is unrelated to independence; independence can exist even if the events occur at different times. The statement about the sum of probabilities equaling 1 relates to the axioms of probability rather than to independence itself. The notion of at least one event always occurring pertains to certain types of events but does not define independent events. Thus, the definition of independence focuses specifically on the lack of influence between events.