FA17:Lecture 32 Prep

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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.

Please come to class prepared with these definitions:

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.