Final answer:
To infer mutual exclusivity from the probabilities P(A) and P(BC), we must verify that P(A AND BC) = 0, indicating that events A and BC cannot occur at the same time. If this probability is not zero, we cannot assume mutual exclusivity without additional information.
Step-by-step explanation:
To infer mutual exclusivity from probabilities such as P(A) and P(BC), we need to understand the definition of mutually exclusive events. Mutually exclusive events are those events that cannot occur at the same time. For example, the probability of event A occurring simultaneously with event B (denoted as P(A AND B)) will be zero if events A and B are mutually exclusive.
An important property of mutually exclusive events is that the probability of one event occurring does not affect the probability of the other. Therefore, if we know that P(A AND C) = 0 (because A and C do not have any numbers in common), then events A and C are mutually exclusive. Another related concept is that if events A and B are mutually exclusive, then the probability of A or B occurring is the sum of their individual probabilities, which means P(A OR B) = P(A) + P(B).
To use these definitions in practice:
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- If you are given P(A) and P(BC), and you want to check for mutual exclusivity, you should attempt to calculate P(A AND BC). If you find that P(A AND BC) = 0, you can confidently conclude that events A and BC are mutually exclusive.
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- If there is no information given that directly shows P(A AND BC) is zero, do not assume that A and BC are mutually exclusive; you need further information or calculation to confirm it.