Sample Space All possible returns on the S&P500 index tomorrow, rounded to the nearest percent –...-4, -3, -2,..., 3, 4,... – This is called sample space, S All possible increases in the index – 1, 2, 3,... – Subset of the sample space, E If return tomorrow is positive, then we’ll say the event has occurred – The outcome is part of E

Events Union of two events: Intersection of two events: Complement of an event: E is contained in F, or E is a subset of F: F is a superset of E:

Rules Commutative Law: Associative Law: Distributive Law:

Rules Continued De Morgan’s laws:

Axioms of Probability Consider an experiment repeated n times – n(E) = number of times an event occurs – Probability of an event = limiting frequency of an event: Or, only assume that P(E) exists

Axioms of Probability Probability of an event E: P(E) Axiom 1: Axiom 2: Axiom 3 *for any sequence of mutually exclusive events

Axioms of Probability What is the probability of a positive return on the S&P500 tomorrow? P({1,2,3,…})= P({1}) + P({2}) + P({3}) + …

Propositions Proposition 4.1. Proposition 4.2. Proposition 4.3.

Equally likely outcomes Imagine all outcomes of a probability space are equal – All outcomes are equally likely

Counting If outcomes are equally likely, all you need to do is count the events in E and S Examples – Two dice, what’s the probability of getting a 7 – Poker hand (5 cards) has 4 of a kind? What’s this?

Counting - Permutatinos How many different ordered arrangements of the letters a, b, and c are possible? n(n-1)(n-2) * 2 * 1 = n! How many different arrangements can be formed from the letters PEPPER? – Example 3d, p. 4

Counting - Combinations Groups of r objects that could be formed from a total of n objects How many groups of 3 could be selected from A, B, C, D, and F?

Probability and Beliefs I’m 60% certain that the market is going down tomorrow.... – And 40% certain that they’re going up the day after tomorrow What’s the probability that – Markets go up both days – Go down both days