Suppose a person is chosen at random. Use your understanding about football to decide whether the event that the person weighs more than 300 pounds and the event that the person plays professional football are independent or associated.

Question

Suppose a person is chosen at random. Use your understanding about football to decide whether the event that the person weighs more than 300 pounds and the event that the person plays professional football are independent or associated.

BlackTom AI · Accepted Answer

We start by restating the scenario to keep the context clear: we’re looking at two events for a randomly chosen person—A = weighs more than 300 pounds, and B = plays professional football—and we want to decide whether A and B are independent or associated.
Option 1: The two events are independent because they are mutually exclusive events. This is incorrect. Mutually exclusive means the events cannot happen at the same time, so P(A ∩ B) = 0. But it is possible for someone to weigh over 300 pounds and also play professional football, so the events are not mutually exclusive. Therefore this justification for independence is flawed.
Option 2: The two events are independent because professional football players are just as likely to weigh more than 300 pounds than the average person. This claim would imply P(A|B) = P(A). While that equality would indicate independence, the statement that players are “just as likely” to weigh >300 pounds as the average person is questionable and counterintuitive given typical player physiques. Moreover, independence is a mathematical condition that requires equality of conditional and marginal probabilities, not a general claim about likelihood without data. As a result, this reasoning is not a solid justification for independence here.
Option 3: The two events are independent because the outcome of one does not affect the outcome of the other. Here a crucial point: independence means exactly that P(A|B) = P(A). However, in the context of football, being a professional player is strongly related to body size and weight, so the occurrence of B likely changes the probability of A (heavier players are more common among professional football players than in the general population). Thus the rationale that one event’s outcome does not affect the other is incorrect for this scenario.
Option 4: The two events are associated because professional football players are more likely to weigh more than 300 pounds than the average person. This aligns with real-world expectations: football players, especially in certain positions, tend to have greater body mass, so P(A|B) > P(A), indicating a positive association. This reasoning directly describes the dependence between the events rather than independence.
In summary, options 1, 2, and 3 each rely on either incorrect assumptions about mutual exclusivity, questionable claim about equal likelihood without data, or a mischaracterization of independence in this context. Option 4 correctly identifies the association based on the plausible link between playing professional football and higher body weight.

Suppose a person is chosen at random. Use your understanding about football to decide whether the event that the person weighs more than 300 pounds and the event that the person plays professional football are independent or associated. 单项选择题

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