FA17:Lecture 32 Prep

In this lecture we will discuss conditional probability, as well as apply this to probability trees and look at some useful tools such as Bayes' rule.

• Last semester's notes (similar to these notes other than the example at the end)

Please come to class prepared with these definitions:

Definition: probability measure

A probability measure on a sample space S is a function [math]\href{/cs2800/wiki/index.php/Pr}{Pr} : \href{/cs2800/wiki/index.php/2}{2}^\href{/cs2800/wiki/index.php/S}{S} \href{/cs2800/wiki/index.php/%E2%86%92}{→} \href{/cs2800/wiki/index.php?title=%E2%84%9D&action=edit&redlink=1}{ℝ} [/math] satisfying the following three properties:

1. For all events [math]E [/math], [math]\href{/cs2800/wiki/index.php/Pr}{Pr}(E) ≥ 0 [/math]
2. [math]\href{/cs2800/wiki/index.php/Pr}{Pr}(S) = 1 [/math]
3. For all disjoint events [math]E_1 [/math] and [math]E_2 [/math], [math]\href{/cs2800/wiki/index.php/Pr}{Pr}(E_1 \href{/cs2800/wiki/index.php/%E2%88%AA}{∪} E_2) = \href{/cs2800/wiki/index.php/Pr}{Pr}(E_1) + \href{/cs2800/wiki/index.php/Pr}{Pr}(E_2) [/math]

These three properties are referred to as the Kolmogorov axioms.

Definition: Probability space

A probability space is just a pair containing a sample space S and a probability measure [math]\href{/cs2800/wiki/index.php/Pr}{Pr} [/math] on S.