Chapter 2 Notes Math 309 Probability

Some Definitions Experiment - means of making an observation Sample Space (S) - set of all outcomes of an experiment listed in a mutually exclusive and exhaustive manner Event - subset of a sample space Simple Event - an event which can only happen in one way; (or can be thought of as a sample point - a one element subset of S)

Since events are sets, we need to understand the basic set operations Intersection (A B) - everything in A and B Union (A B) - everything in A or B or both Complement ( ) - everything not in A

You should be able to sketch Venn diagrams to describe the intersections, unions, & complements of sets. Note that these set operations obey the commutative, associative, and distributive laws

DeMorgan’s Laws Convince yourself that these are reasonable with Venn diagrams!

Axioms of Probability (these are FACT, no proof needed!) Let A represent an event, S the sample space, P(A) 0 P(S) = 1 For pairwise mutually exclusive events, the probability of their union is the sum of their respective probabilities, i.e.

Propositions (You should be able to prove these using the axioms and definitions.) Let E and F be any two events. 2.1 0 P(E) 1 2.2 If E is a subset of F, then P(E) P(F). 2.3 For mutually exclusive events, P(E F) = P(E) + P(F)

Theorems (You should be able to prove these using the axioms and definitions.) Let E and F be any two events. Thm 2.5 P( ) = 1 - P(E) Thm 2.6 P(E F) = P(E) + P(F) - P(E F)

Sample Spaces with Equally Likely Outcomes In an experiment where all simple events (sample points) are equally likely, one can find the probability of an event by counting two sets.

Unions get complicated if events are not mutually exclusive! P(A  B  C) = P(A) + P(B) + P(C) - P(A  B) - P(A  C) - P(B  C) + P(A  B  C) B