21. As part of an exercise regimen, the probability of a person running outside is 0.45, the probability of a person joining a gym is 0.75, and the probability of a person both running outside and joining a gym is 0.25. Which conclusion can be drawn about these three events?
Answer: C
The events are dependent.
The probabilities provided indicate that the outcomes of running outside and joining a gym do not operate independently of one another. Specifically, the overlap in probabilities suggests a dependency between the two activities.
A) The events are not intersecting.
This option is incorrect because the events do intersect; the probability of both running outside and joining a gym is 0.25, indicating that there is an overlap between the two activities.
B) The events are not disjoint.
While this statement is true, it does not fully capture the nature of the relationship between the events. The events being not disjoint suggests they can occur simultaneously, but it does not address the dependency implied by the given probabilities.
C) The events are dependent.
This is the correct conclusion, as the probability of a person participating in one activity affects the probability of participating in the other. The fact that both activities have a joint probability indicates that knowing one event influences the likelihood of the other occurring.
D) The events are equally distributed.
This option is incorrect as it implies that both activities have the same probability of occurrence, which is not the case here. The probabilities of running outside (0.45) and joining a gym (0.75) are not equal, indicating a lack of equal distribution.
Conclusion
The conclusion that the events are dependent is strongly supported by the probabilities provided. All other options fail because they either misinterpret the nature of the intersection or do not adequately express the relationship between the two events. Thus, the dependency of these activities is the most accurate representation of their interaction.