The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 171 Wednesday, February 4, 2004 3:00 PM

Concepts of Probability 171

PA ( ∩B)

B)= ------------------------

PB

PA(

()

It is possible to deduct from simple properties of set theory and from the

disjoint additivity of probability that

P(A ∪B) = P(A) + P(B) – P(A ∩B) ≤P(A) + P(B)

P(A) = 1 – P(AC)

Bayes theorem is a rule that links conditional probabilities. It can be

stated in the following way:

PA ( ∩B) PA ( ∩B)PA() PA()

PA( ( A)-------------

() PB ()

B) = ------------------------ = -------------------------------------- = PB

PB ()PA PB()

Bayes theorem allows one to recover the probability of the event A

given B from the probability of the individual events A, B, and the prob-

ability of B given A.

Discrete probabilities are a special instance of probabilities. Defined

over a finite or countable set of outcomes, discrete probabilities are non-

zero over each outcome. The probability of an event is the sum of the

probabilities of its outcomes. In the finite case, discrete probabilities are

the usual combinatorial probabilities.

MEASURE

A measure is a set function defined over an algebra or σ-algebra of sets,

denumerably additive, and such that it takes value zero on the empty set

but can otherwise assume any positive value including, conventionally,

an infinite value. A probability is thus a measure of total mass 1 (i.e., it

takes value 1 on the set Ω).

A measure can be formally defined as a function M(A) from an alge-

bra or a σ-algebra ℑto R (the set of real numbers) that satisfies the fol-

lowing three properties:

■ Property 1. 0 ≤M(A), for every A ∈ ℑ

■ Property 2. M(∅) = 0