Understanding Event Independence

Two events, A and B, are considered independent if the occurrence of one event does not affect the probability of the other event occurring. Mathematically, this relationship is defined by specific probability conditions.

Conditions for Independence

The primary condition that defines the independence of two events A and B is when the joint probability, denoted as $P(A, B)$ or $P(A \cap B)$, is equal to the product of their individual probabilities:

$ P(A \cap B) = P(A) P(B) $

Other equivalent conditions derived from this definition include:

• If $ P(B) > 0 $, then $ P(A|B) = P(A) $.
• If $ P(A) > 0 $, then $ P(B|A) = P(B) $.

Here, $ P(A|B) $ represents the conditional probability of A occurring given that B has occurred, and $ P(B|A) $ represents the conditional probability of B occurring given that A has occurred.

Analysis of Options

Let's examine each option in the context of event independence:

• Option 1: $ P(A/B) = P(A) $: This condition is satisfied if events A and B are independent (assuming $ P(B) > 0 $). It implies that the probability of A occurring is unaffected by the knowledge that B has occurred.
• Option 2: $ P(B/A) = P(B) $: Similarly, this condition is satisfied if events A and B are independent (assuming $ P(A) > 0 $). It means the probability of B is not changed by the occurrence of A.
• Option 3: $ P(A, B) = P(A) P(B) $: This is the fundamental definition of statistical independence for two events. It states that the probability of both A and B occurring together is the product of their individual probabilities.
• Option 4: $ P(A, B) = P(A) + P(B) $: This equation does NOT satisfy the condition for independence. In probability theory, $ P(A) + P(B) $ is related to the probability of the union of two events ($ P(A \cup B) = P(A) + P(B) - P(A \cap B) $). If $ P(A, B) = P(A) + P(B) $, it would imply $ P(A \cap B) = P(A) + P(B) $. Substituting the independence condition $ P(A \cap B) = P(A)P(B) $, we get $ P(A)P(B) = P(A) + P(B) $. This equation is generally not true for independent events unless specific trivial cases apply (like $ P(A)=0 $ or $ P(B)=0 $). It is characteristic of mutually exclusive events' union, not independence.

Conclusion

The condition that is not satisfied for the independence of two events A and B is $ P(A, B) = P(A) + P(B) $. This equation relates to the probability of the union of mutually exclusive events, not the joint probability of independent events.