Problem 72
Question
Determine whether the statement is true or false. Justify your answer. If \(A\) and \(B\) are independent events with nonzero probabilities, then \(A\) can occur when \(B\) occurs.
Step-by-Step Solution
Verified Answer
The statement is true. If \(A\) and \(B\) are independent events, then \(A\) can occur when \(B\) occurs because the occurrence of one event does not affect the occurrence of the other.
1Step 1: Understanding Independence
Two events are said to be independent if the occurrence of one event does not influence the occurrence of the other. In terms of probability, this can be written as \(P(A \cap B) = P(A) \cdot P(B)\), where \(A\) and \(B\) are two independent events.
2Step 2: Application on Given Problem
Given \(A\) and \(B\) are independent events with nonzero probabilities, it means that the occurrence of \(A\) does not affect the occurrence of \(B\) and vice versa.
3Step 3: Decision on the Statement
Therefore, \(A\) can occur when \(B\) occurs because the occurrence of \(B\) does not affect the occurrence of \(A\). Hence, the original statement is true.
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