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This exercise describes an example of a nonmeasurable set in R. If x and y are real numbers in the interval [0, 11, we say that x and y are equivalent, written x - y is rational. The relation is an equivalence relation, and the interval [0, 11] can be expressed as a disjoint union of subsets (called equivalence classes) in each of which no two distinct points are equivalent. Choose a point from each equivalence class and let E be the set of points so chosen. We assume that E is measurable and obtain a contradiction. Let denote the set of rational numbers in [-1, 11 and let . a) Prove that each is measurable and that . b) Prove that s a disjoint collection of sets whose union contains [0, 1] and is contained in [-1, 2]. c) Use parts (a) and (b) along with the countable additivity of Lebesgue measure to obtain a contradiction.
If one event can occur in ways AND a second event can occur in ways, then the number of ways that the two events can occur in sequence is then .
There are 3 options for each of the 10 people (Coke, Pepsi, and RC) and thus there are possible outcomes by the multiplication principle.
The order in which we select the Coke answers does not matter (as a different order results in the same people answering Coke) and thus we need to use the definition of .
There are ways to select 0 of the 10 people (who answer Coke), thus there are ways to select at least one person how answer Coke and thus there are favorable outcomes.
The probability is the number of favorable outcomes divided by the number of possible outcomes:
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