A Primer on probability for design inferences

Abstract
Probabilities are numbers between 0 and 1 that attach to events. Events always occur with respect to a reference class of possibilities. Consider a die with faces 1 through 6. The reference class of possibilities in this case can be represented by the set {1, 2, 3, 4, 5, 6}. Any subset of this reference class then represents an event. For instance, the event Eodd, i.e., “an odd number was tossed,” corresponds to {1, 3, 5}. Such an event is said occur if any one of its outcomes occurs, i.e., if either a 1 is tossed or a 3 or a 5. Outcomes can therefore be represented as singleton sets, i.e., sets with only one element. Thus, the outcomes associated with Eodd = {1, 3, 5} are E1 = {1}, E3 = {3}, and E5 = {5}. Outcomes are sometimes also called elementary events. Events include not only outcomes but also composite events like Eodd that include more than one outcome.
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