Which statement correctly relates conditional probability and independence?

Which statement correctly relates conditional probability and independence?

• A. If A and B are independent, then P(A | B) = P(A)
• B. If A and B are dependent, then P(A | B) = 0
• C. P(A | B) is always greater than P(A)
• D. Independence means P(A ∩ B) is always zero

Answer: (A)

The correct statement is that if events A and B are independent, then the conditional probability of A given B, denoted as P(A | B), is equal to the probability of A, or P(A). This scenario reflects the nature of independence in probability theory, which states that the occurrence of one event does not affect the occurrence of the other. Mathematically, this relationship can be expressed as: \[ P(A | B) = \frac{P(A \cap B)}{P(B)} \] Since the events are independent, the joint probability P(A ∩ B) can be expressed as the product of their individual probabilities, P(A) * P(B). This implies that: \[ P(A | B) = \frac{P(A) \cdot P(B)}{P(B)} = P(A) \] This reinforces the concept that the knowledge of event B occurring does not change the probability of event A occurring, confirming that their relationship is independent. Other statements don't accurately describe the relationship between conditional probability and independence. For example, saying that if A and B are dependent then P(A | B) = 0 would only be true in a very specific case where knowing B guarantees that A cannot happen, which is

The correct statement is that if events A and B are independent, then the conditional probability of A given B, denoted as P(A | B), is equal to the probability of A, or P(A). This scenario reflects the nature of independence in probability theory, which states that the occurrence of one event does not affect the occurrence of the other. Mathematically, this relationship can be expressed as:

[ P(A | B) = \frac{P(A \cap B)}{P(B)} ]

Since the events are independent, the joint probability P(A ∩ B) can be expressed as the product of their individual probabilities, P(A) * P(B). This implies that:

[ P(A | B) = \frac{P(A) \cdot P(B)}{P(B)} = P(A) ]

This reinforces the concept that the knowledge of event B occurring does not change the probability of event A occurring, confirming that their relationship is independent.

Other statements don't accurately describe the relationship between conditional probability and independence. For example, saying that if A and B are dependent then P(A | B) = 0 would only be true in a very specific case where knowing B guarantees that A cannot happen, which is