Question at position 3 Based on your calculations, is the occurrence of the nos gene independent of being a centenarian?No because the product of the marginal probabilities is equal to the probability of their intersectionNo because the product of the marginal probabilities is not equal to the probability of their intersectionYes because the product of the marginal probabilities is equal to the probability of their intersectionYes because the product of the marginal probabilities is not equal to the probability of their intersectionBlackTom题目解析

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Question at position 3 Based on your calculations, is the occurrence of the nos gene independent of being a centenarian?No because the product of the marginal probabilities is equal to the probability of their intersectionNo because the product of the marginal probabilities is not equal to the probability of their intersectionYes because the product of the marginal probabilities is equal to the probability of their intersectionYes because the product of the marginal probabilities is not equal to the probability of their intersectionBlackTom题目解析

BlackTom AI · Accepted Answer

First, restating the core idea in play: we’re evaluating whether the nos gene occurring in a person is independent of being a centenarian, using the criterion of independence that relies on probability products.
Option 1: 'No because the product of the marginal probabilities is equal to the probability of their intersection.' If the product of the marginals did equal the intersection probability, that would indicate independence. Here the claim asserts equality, which would support independence. However, the statement as given is a declaration about equality without acknowledging whether that equality actually holds in the data. In many practical problems, equality of marginals to the joint product is not guaranteed, so this option seems to misuse the independence criterion by assuming the equality holds.
Option 2: 'No because the product of the marginal probabilities is not equal to the probability of their intersection.' This aligns with the standard test for independence: if P(A and B) ≠ P(A)P(B), then A and B are not independent. Therefore, this option claims a dependence between nos gene occurrence and being a centenarian, which would indeed justify that they are not independent.
Option 3: 'Yes because the product of the marginal probabilities is equal to the probability of their intersection.' This asserts independence by equality of joint and product of marginals. As with Option 1, the correctness hinges on actual data. If the equality does not hold, this option is incorrect. Since the given correct choice states that the product is not equal, this option would be false in this scenario.
Option 4: 'Yes because the product of the marginal probabilities is not equal to the probability of their intersection.' This mixes up the logic: independence is about equality (P(A and B) = P(A)P(B)). If the product is not equal, that indicates dependence, so labeling it as 'Yes' for independence is incorrect. This option contradicts the formal criterion for independence.
To summarize the reasoning for each choice, we’re scrutinizing the independence condition P(A ∩ B) = P(A)P(B). When the product of marginals does not equal the joint probability, A and B are not independent; that is the standard cue for dependence rather than independence. In this problem, the statement that the product is not equal to the intersection probability correctly indicates dependence, which rules out independence.
Hence, the option asserting the non-equality (Option 2) correctly identifies the dependence between the nos gene occurrence and being a centenarian, while the options claiming independence or equality (Options 1 and 3) are incorrect in this context, and the final option incorrectly combines a non-equality with an affirmation of independence (Option 4).

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