University of California, San Diego: Edward Bender and S. Williamson's "Lists, Decisions, and Graphs: Counting and Listing"
Read Section 4 on pages CL-28 through CL-30.
Some useful results are summarized here:
∑xε S P(x) = 1, where S is the sample space, also written P(S) = 1;
P(x) >=0, for x in the sample space;
Let E be an event, defined as a subset of S. P(E) = ∑xε EP(x);
If E and D are events such that E is contained in D, then P(E) <= P(D); and
P(E È D) = P(E) + P(D) for E, D disjoint.
From the above the following can be proved: Let E be an event. E' = S -E. P(E È E') = P(E) + P(E'), P(S= E È E') = 1; thus, 1 = P(E) + P(E') and P(E') = 1 - P(E).
Note: the probability of the complement of an event E is 1 - (probability of E). This result is often very useful, because for some problems the probability of E may be difficult to calculate, but the probability of the complement of E may be easy to calculate.