Which condition guarantees independence of A and B?

Which condition guarantees independence of A and B?

• A. A and B are independent if P(A∩B) = P(A)P(B)
• B. A and B are independent because P(A|B) = P(A)
• C. A and B are never independent
• D. A and B are independent if P(A|B) ≠ P(A)

Answer: (A)

Independence means the occurrence of one event does not change the probability of the other. The clean, defining way to express that is the product rule: P(A and B) equals P(A) times P(B). If this holds, knowing that B occurred doesn’t change how likely A is, and vice versa. P(A|B) = P(A) is another way to state independence, but it relies on the conditional probability being defined (P(B) > 0). The product form is more universally applicable and doesn’t require that extra condition. The idea that they are never independent is false, since many event pairs are independent. Saying P(A|B) ≠ P(A) suggests dependence, not independence.

Independence means the occurrence of one event does not change the probability of the other. The clean, defining way to express that is the product rule: P(A and B) equals P(A) times P(B). If this holds, knowing that B occurred doesn’t change how likely A is, and vice versa.

P(A|B) = P(A) is another way to state independence, but it relies on the conditional probability being defined (P(B) > 0). The product form is more universally applicable and doesn’t require that extra condition. The idea that they are never independent is false, since many event pairs are independent. Saying P(A|B) ≠ P(A) suggests dependence, not independence.