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\title{Foundations of Probability\footnote{ This slide show is an open-source document. See last slide for copyright information.}
 \\ (Sections 1.2 and 1.3 in the text)}
\subtitle{STA 256: Fall 2019}
\date{} % To suppress date

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  \titlepage
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\begin{frame}
\frametitle{The idea}
%\framesubtitle{} 
{\Large
Informally, a probability is a number between zero and one indicating how likely an event is to occur.
} \pause \vspace{5mm}
\begin{itemize}
     \item Sample space $S$ is the set of all things that can happen. \pause
     \item Elements $s \in S$ are called \emph{outcomes}. \pause
     \item Subsets $A \subseteq S$ are called \emph{events}.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Basic Properties of Probability}
\framesubtitle{p.~5 in text}  \pause
A probability measure is a function $P$ from subsets of $S$ to the real numbers, satisfying \pause
\begin{enumerate}
    \item $0 \leq P(A) \leq 1$ \pause
    \item $P(\emptyset) = 0$ \pause
    \item  $P(S)=1$ \pause
    \item  If $A_1, A_2 \ldots$ are disjoint subsets of $S$, \pause
          $P\left( \cup_{k=1}^\infty A_k \right) = \sum_{k=1}^\infty P(A_k)$.
\end{enumerate}
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\begin{frame}
\frametitle{A Boring Example} \pause
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Sell 500 lottery tickets, pick the winning number.  \pause

\begin{itemize}
    \item $S = \{1, 2, \ldots, 500 \}$ \pause
    \item $P(\{2\}) = 1/500$ \pause
    \item $P(\mbox{Even Number}) = 1/2$
\end{itemize}
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\begin{frame}
\frametitle{``Basic Properties" are really axioms (Kolmogorov, 1933)}
\framesubtitle{The properties are a little redundant} \pause 
\begin{columns}  
\column{0.5\textwidth} 
\begin{center} Basic Properties \end{center}
  \begin{enumerate}
    \item $0 \leq P(A) \leq 1$ 
    \item $P(\emptyset) = 0$ 
    \item  $P(S)=1$ \pause
    \item  If $A_1, A_2 \ldots$ are disjoint subsets of $S$, 
          $P\left( \cup_{k=1}^\infty A_k \right) = \sum_{k=1}^\infty P(A_k)$.
  \end{enumerate} \pause
\column{0.5\textwidth}
\begin{center} Axioms \end{center}
\begin{enumerate}
    \item $P(A) \geq 0$ for any $A \subseteq S$ 
    \item $P(S)=1$ 
    \item If $A_1, A_2 \ldots$ are disjoint subsets of $S$,
          $P\left( \cup_{k=1}^\infty A_k \right) = \sum_{k=1}^\infty P(A_k)$ 
\end{enumerate}
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\frametitle{Starting from the axioms, one can show} \pause
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\begin{itemize}
    \item $P(\emptyset) = 0$.  \pause
    \item If $A_1, \ldots, A_n$ are disjoint,  $P\left( \cup_{k=1}^n A_k \right) = \sum_{k=1}^n P(A_k)$ \pause (finite additivity). \pause
    \item Then it's smooth sailing. \pause
    \item In this course, we will start with the 4 properties, and assume that Property~4 (additivity) applies to either finite or infinite collections of sets.
\end{itemize}
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\frametitle{Some Elementary Theorems}  \pause % Need to display properties on another screen for proofs
% \framesubtitle{}
\begin{itemize}
    \item $P(A^c) = 1-P(A)$ \pause
    \item If $A \subseteq B$ then $P(A) \leq P(B)$ \pause
    \item $P(A \cup B) = P(A)+P(B)-P(A\cap B)$ \pause (Inclusion-exclusion principle)
\end{itemize}
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\begin{frame}  
\frametitle{Some Not-So-Elementary Theorems}  \pause
% \framesubtitle{}
\begin{itemize}
    \item Law of total Probability\pause: Let $A_1, A_2, \ldots$ form a partition of the sample space $S$, and let $B$ be any event. Then $P(B) = \sum_{k=1}^\infty (A_k \cap B)$. \pause
    \item Sub-additivity\pause: Let $A_1, A_2, \ldots$ be a collection of events, not necessarily disjoint. Then \\
    $P\left( \cup_{k=1}^\infty A_k\right) \leq \sum_{k=1}^\infty P(A_k) $ \pause
    \item Continuity 1\pause: Let $A_1 \subseteq A_2 \subseteq A_3 \subseteq \ldots$ and let $A = \cup_{k=1}^\infty A_k$. \pause Then $\lim_{k \rightarrow \infty} P(A_k) = P(A)$. \pause
    \item Continuity 2\pause: Let $A_1 \supseteq A_2 \supseteq A_3 \supseteq \ldots$ and let $A = \cap_{k=1}^\infty A_k$. \pause Then $\lim_{k \rightarrow \infty} P(A_k) = P(A)$.
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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:

\vspace{5mm}
\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f19} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f19}}

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

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    \item  \pause
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    \item 
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\begin{comment}
\begin{frame}
\frametitle{Axioms of Probability}
\framesubtitle{Kolmogorov (1933)}  \pause
A probability measure is a function $P$ from subsets of $S$ to the real numbers, satisfying \pause
\begin{enumerate}
    \item $P(S)=1$ \pause
    \item $P(A) \geq 0$ for any $A \subset S$ \pause
    \item If $A_i \cap A_j = \emptyset$ for $A_1, A_2 \ldots$, where $i \neq j$, \pause
          $P\left( \cup_{k=1}^\infty A_k \right) = \sum_{k=1}^\infty P(A_k)$ 
\end{enumerate}
\end{frame}
\end{comment}

Axioms in LaTeXit (36 point) for acetate

\noindent Fundamental Properties of Probability \vspace{5mm}

\noindent
A probability measure is a function $P$ from subsets of $S$  \\ to the real numbers, satisfying 
\begin{enumerate}
    \item $0 \leq P(A) \leq 1$ \pause
    \item $P(\emptyset) = 0$ \pause
    \item  $P(S)=1$ \pause
    \item  If $A_1, A_2 \ldots$ are disjoint subsets of $S$, \pause
          $P\left( \cup_{k=1}^\infty A_k \right) = \sum_{k=1}^\infty P(A_k)$.
\end{enumerate}