Math 207 – Introduction to Probability and Statistics

Proofs of Probability Propositions (additional, required course material)

Axioms of Probability

1. , for any event A
2. , where S is the sample space
3. For mutually exclusive events

Propositions Following from the Axioms

1. 

Proof:

First note that A and are mutually exclusive for any event A. Then, since we have by Axioms 2 and 3 that . Hence (by simple algebra), .

Corresponding Venn Diagram: Diagram not shown with online handout (but drawn on handout given in class)

2. If A is a subset of B, then

Proof:

Since A is a subset of B, it follows that we can express B as Furthermore, A and are mutually exclusive, so from Axiom 3 we know , since Hence,

Corresponding Venn Diagram: Diagram not shown with online handout (but drawn on handout given in class)

3. 

Proof:

To derive a formula for first note that can be written as the union of the two mutually exclusive events A and Thus from Axiom 3 we obtain Furthermore, since we again obtain from Axiom 3 that or, equivalently, Then, by substitution, .

Corresponding Venn Diagrams: Diagram not shown with online handout (but drawn on handout given in class)