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:

FE

FE

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).... 3 * 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