The probability that a given urine sample analyzed in a pathology lab contains protein is 0.73 (event A). The probability that the urine sample contains the protein albumin is 0.58 (event B).
- Event A: The sample contains protein.
- Event B: The sample contains the protein albumin.
The probability that a sample contains protein and that the protein is albumin is 0.58. Which statement is true?
A. Events A and B are dependent because \( P(A \text{ and } B) \neq P(A) + P(B) \).
B. Events A and B are dependent because \( P(A \text{ and } B) = P(A) + P(B) \).
C. Events A and B are dependent because \( P(A \text{ and } B) = P(A)P(B) \).
D. Events A and B are dependent because \( P(A \text{ and } B) \neq P(A)P(B) \).
Answer (2)
E v e n t A an d e v e n t B a re in d e p e n d e n t ⇔ P ( A ∩ B ) = P ( A ) ⋅ P ( B ) − − − − − − − − − − − − − − − − − − − − − P ( A ) = 0.78 an d P ( B ) = 0.58 an d P ( A ∩ B ) = 0.58 P ( A ) ⋅ P ( B ) = 0.78 ⋅ 0.58 = 0.58 ⇒ E v e n t A an d e v e n t B a re d e p e n d e n t ⇒ A n s . ( ?)
Events A and B are dependent because the probability of both events occurring together is not equal to the product of their individual probabilities. Thus, option D is the correct answer. This shows that knowing one event provides information about the other event's likelihood.
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