Set theory
Definition: the set, \(S\) , of all possible outcomes of a particular experiment is called the sample space for the experiment
i.e.
- flipping a coin
\(S = \{H,T\}\)
- reaction time to a stimulus
\(S = (0, \infty )\)
Definition: an event is any collection of possible outcomes of an experiments, that is, any subset of \(S\) (include \(S\) itself)
-
Union is the set of elements that belong to either A or B or both \(A \cup B = \{x:x \in A or x \in B \}\)
-
Intersection is the set of elements that belong to both A and B \(A \cap B = \{x:x \in A or x \in B \}\)
-
Complementation is the set of all elements that are not in A \(A^c = \{ x:x \notin A\}\)
Theorem: for any three events, A, B, and C, defined on a sample space S
a. Commutativity b. Associativity c. Distributive Laws d. DeMorgan's Laws
Definition: Two events A and B are disjoint (or mutually exclusive) if \(A \cap B = \emptyset\). The events $A1, A2, …$ are pairwise disjoint (or mutually exclusive) if \(A_i \cap A_j = \emptyset\) for all \(i \neq j\)
Definition: If \(A_1, A_2\) … are pairwise disjoint and \(\cup_{i=1}^\infty A_i = S\), then the collection \(A_1, A_2\) … forms a partition of \(S\)