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\title{Foundations of Probability\footnote{ This slide show is an open-source document. See last slide for copyright information.}}
\subtitle{STA 256: Fall 2018}
\date{} % To suppress date

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\frametitle{The idea}
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{\Large
Informally, a probability is a number between zero and one indicating how likely an event is to occur.
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\frametitle{Axioms of Probability}
\framesubtitle{Kolmogorov (1933)}  \pause
A probability measure is a function $P$ from subsets of $\Omega$ to the real numbers, satisfying \pause
\begin{enumerate}
    \item $P(\Omega)=1$ \pause
    \item $P(A) \geq 0$ for any $A \subset \Omega$ \pause
    \item If $A_i \cap A_j = \emptyset$ for $A_1, A_2 \ldots$, where $i \neq j$, \pause
          $P\left( \cup_{k=1}^\infty A_k \right) = \sum_{k=1}^\infty P(A_k)$ 
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\frametitle{Properties of Probability} \pause
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    \item[A.] $P(A^c) = 1-P(A)$ \pause
    \item[B.] $P(\emptyset)=0$ \pause
    \item[C.] If $A \subseteq B$ then $P(A) \leq P(B)$ \pause
    \item[D.] $P(A \cup B) = P(A)+P(B)-P(A\cap B)$ \pause (The addition law)
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\frametitle{Copyright Information}

This slide show was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistical Sciences, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website:

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\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f18} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f18}}

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$ = \{\omega \in \Omega: \}$

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