Final answer:
The question involves applying mathematics, particularly probability rules, to compute event likelihood and judge event dependence and exclusivity in statistics. It requires knowledge of the multiplication and addition rules and understanding of sampling methods, independent and dependent events, and mutually exclusive scenarios.
Step-by-step explanation:
The question pertains to probability, a branch of mathematics that deals with calculating the likelihood of various events occurring in different situations. Specifically, it involves understanding and applying two fundamental rules: the multiplication rule and the addition rule of probability, and assessing whether events are independent or dependent. These rules are essential for determining the probability of events occurring together ('A and B') using the multiplication rule, and the probability of at least one event occurring ('A or B') using the addition rule.
In the context of sampling with replacement, where each member of a population can be chosen more than once, events are considered independent, since the outcome of one does not affect the outcome of another. In contrast, sampling without replacement implies that once a member is picked, it cannot be picked again, making events dependent. Furthermore, understanding mutually exclusive events, which do not share any outcomes, is crucial for applying the addition rule correctly.
For independent events, the probability of both events A and B occurring can be found by multiplying the probabilities of the individual events (P(A and B) = P(A)P(B)). When events are not independent, conditional probability and other methods come into play for accurate calculations.