To determine which statement about the independence of events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is true, we need to understand the condition that characterizes independent events. The events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are considered independent if and only if
[tex]\[ P(A \mid B) = P(A). \][/tex]
We have four statements to evaluate:
1. [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent events because [tex]\( P(A \mid B) = P(A) = 0.18 \)[/tex].
2. [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent events because [tex]\( P(A \mid B) = P(A) = 0.4 \)[/tex].
3. [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are not independent events because [tex]\( P(A \mid B) = 0.4 \)[/tex] and [tex]\( P(A) = 0.18 \)[/tex].
4. [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are not independent events because [tex]\( P(A \mid B) = 0.18 \)[/tex] and [tex]\( P(A) = 0.4 \)[/tex].
Let's analyze each statement one by one:
1. For the statement to be true, [tex]\( P(A \mid B) \)[/tex] must equal [tex]\( P(A) \)[/tex], and here both are given as 0.18. This condition satisfies the definition of independence. Therefore, this statement is true.
2. For this statement, [tex]\( P(A \mid B) \)[/tex] is given as 0.4, and [tex]\( P(A) \)[/tex] is also given as 0.4. Again, [tex]\( P(A \mid B) = P(A) \)[/tex] matches the definition of independence, so this statement is also true under the given condition.
3. This statement is evaluating dependence because it states [tex]\( P(A \mid B) = 0.4 \)[/tex] while [tex]\( P(A) = 0.18 \)[/tex]. Since 0.4 does not equal 0.18, the events are not independent. However, this statement does not align with the conditions that confirm the presence of independence.
4. Similarly, this statement is evaluating the mismatch wherein [tex]\( P(A \mid B) = 0.18 \)[/tex] whereas [tex]\( P(A) = 0.4 \)[/tex]. Clearly, 0.18 does not equal 0.4, indicating dependence. But again, this does not match the given conditions.
Given these conditions, the only valid supporting evidence confirming independence without any further mismatch is:
[tex]\[ \text{Statement 1: } A \text{ and } B \text{ are independent events because } P(A \mid B) = P(A) = 0.18. \][/tex]
Thus, the correct statement is:
[tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent events because [tex]\( P(A \mid B) = P(A) = 0.18 \)[/tex].
