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cs-401r:axioms-of-probability-theory [2014/09/05 16:54]
cs401rpml [Second Axiom]
cs-401r:axioms-of-probability-theory [2014/09/05 17:02]
cs401rpml [Second Axiom]
Line 1: Line 1:
 == First Axiom ==  == First Axiom == 
 The probability of any event $E$ is between 0 and 1: The probability of any event $E$ is between 0 and 1:
-\begin{equation} + 
-0 \leq P\left(E\right) \leq 1 +$0 \leq P\left(E\right) \leq 1$
-\end{equation}+
  
 == Second Axiom ==  == Second Axiom == 
 The probability of the entire sample space $\Omega$ (equivalently,​ the "​certain event"​) is 1. The probability of the entire sample space $\Omega$ (equivalently,​ the "​certain event"​) is 1.
-\begin{equation} + 
-P\left(\Omega\right) = 1 +$P\left(\Omega\right) = 1$ 
-\end{equation}+
  
 == Third Axiom ==  == Third Axiom == 
-If for all <​math>​0 \leq i, j</​math> ​such that <​math>​i \neq j</​math> ​the events ​<​math>​A_i</​math> ​and <​math>​A_j</​math> ​are disjoint (i.e., ​<​math>​A_i \cap A_j = \emptyset</​math>​), then +If for all $0 \leq i, jsuch that $i \neq jthe events ​$A_iand $A_jare disjoint (i.e., ​$A_i \cap A_j = \emptyset$), then 
-:<​math>​+\begin{equation}
 P\left(\bigcup_{j=1}^{\infty}{A_j}\right) = \sum_{j=1}^{\infty}{P\left(A_j\right)} P\left(\bigcup_{j=1}^{\infty}{A_j}\right) = \sum_{j=1}^{\infty}{P\left(A_j\right)}
-</​math>​+\end{equation} 
 + 
 +In the simple case of two variables, if $A \cap B = \emptyset$ then: 
 +\begin{equation} 
 +P\left(A \cup B\right) = P\left(A\right) + P\left(B\right) 
 +\end{equation}
  
-In the simple case of two variables, if <​math>​A \cap B = \emptyset</​math>​ then: 
-:<​math>​P\left(A \cup B\right) = P\left(A\right) + P\left(B\right)</​math>​ 
  
cs-401r/axioms-of-probability-theory.txt · Last modified: 2014/09/05 17:02 by cs401rpml
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