To determine the condition under which [tex]\( P(A \text{ or } B) = P(A) + P(B) \)[/tex] is true, we need to consider the nature of events [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
1. Event A and Event B must be the only outcomes that can occur:
This condition is not relevant to whether the equation [tex]\( P(A \text{ or } B) = P(A) + P(B) \)[/tex] holds. We are concerned with the overlap of the events, not the exclusivity of all possible outcomes.
2. Event A and Event B must share at least one outcome in common:
If events share at least one outcome in common, they are not mutually exclusive. This means that if [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are happening, there is some probability overlap that needs to be subtracted to avoid double-counting. Thus, if [tex]\( P(A \)[/tex] and [tex]\( B) \neq \)[/tex] 0, the equation must be adjusted to be [tex]\( P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \)[/tex]. Therefore, this condition does not satisfy the given equation.
3. Event A and Event B must not be able to occur at the same time:
This implies that events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are mutually exclusive. In probability terms, this would be [tex]\( P(A \text{ and } B) = 0 \)[/tex]. When this is the case, the equation [tex]\( P(A \text{ or } B) = P(A) + P(B) \)[/tex] holds true, as there is no overlap between the two events that needs to be subtracted.
Therefore, the correct condition for [tex]\( P(A \text{ or } B) = P(A) + P(B) \)[/tex] to hold true is:
Event A and event B must not be able to occur at the same time.
