# Elementary Sets as Events and Their Complements

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## Chapter: Biostatistics for the Health Sciences: Basic Probability

The elementary events are the building blocks (or atoms) of a probability model.

ELEMENTARY SETS AS EVENTS AND THEIR COMPLEMENTS

The elementary events are the building blocks (or atoms) of a probability model. They are the events that cannot be decomposed further into smaller sets of events. The set of elementary events is just the collection of all the elementary events. In example 2, the event {1, 1} “snake eyes” is an elementary event. The set [{1, 1}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 1}, {2, 2}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 1}, {3, 2}, {3, 3}, {3, 4}, {3, 5}, {3, 6}, {4, 1}, {4, 2}, {4, 3}, {4, 4}, {4, 5}, {4, 6}, {5, 1}, {5, 2}, {5, 3}, {5, 4}, {5, 5}, {5, 6}, {6, 1}, {6, 2}, {6, 3}, {6, 4}, {6, 5}, and {6, 6}] is the set of elementary events.

It is customary to use Ω, the Greek letter omega, to represent the set containing all the elementary events. This set is also called the universal set. For Ω we have P(Ω) = 1. The set containing no events is denoted by Ø and is called the null set, or empty set. For the empty set Ø we have P(Ø) = 0.

For any set A, Ac denotes the complement of A. The complement of set A is just the set of all elementary events not contained in A. From Example 2, if A = {sum of the faces on the two dice is seven}, then A = [{1, 6}, {2, 5}, {3, 4}, {4, 3}, {5, 2}, {6, 1}] and the set Ac is the set [{1, 1}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 1}, {2, 2}, {2, 3}, {2, 4}, {2, 6}, {3, 1}, {3, 2}, {3, 3}, {3, 5}, {3, 6}, {4, 1}, {4, 2}, {4, 4}, {4, 5}, {4, 6}, {5, 1}, {5, 3}, {5, 4}, {5, 5}, {5, 6}, {6, 2}, {6, 3}, {6, 4}, {6, 5}, and {6, 6}] .

By simply counting the elementary events in the set and dividing by the total number of elementary events in Ω, we obtain the probability for the event. In problems with a large number of elementary events, this method for finding a probability can be tedious; it also requires that the elementary events are equally likely. Formulas that we derive in later sections will allow us to compute more easily the probabilities of certain events.

Consider the probability of A = {sum of the faces on the two dice is seven}. As we saw in the previous section, P(A) = 6/36 = 1/6 0.167. Since there are 30 elementary events in Ac, P(Ac) = 30/36 = 5/6 0.833. We see that P(Ac) = 1 – P(A), which is always the case, as demonstrated in the next section.