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\usepackage{amsthm}

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\title{MA162: Finite mathematics}
\subtitle{Introduction to Probability}
\author{Paul Koester}
\institute{University of Kentucky}
\date{February 19, 2014}

\begin{document}

\maketitle

\begin{frame}[t]{7.1: Probability Theory Basics}
 \begin{itemize}
  \item There is much overlap between Counting (Chapter 6) and Probability Theory (Chapter 7), but some terminology is different \vfill
  \item An \alert{experiment} is an activity with observable results  \vfill
  \item A \alert{sample point} is an outcome of an experiment \vfill
  \item A \alert{sample space} is the collection of all sample points of an experiment \vfill
  \item An \alert{event} is a collection of some (possibly all, possibly none) sample points of an experiment \vfill
  \item A \alert{simple event} is an event containing exactly one sample point \vfill 
 \end{itemize}
\end{frame}

\begin{frame}[t]{7.1: Counting to Probability Dictionary}
 \begin{tabular}{lll}
  \textbf{Counting} 	&			& \textbf{Probability} \\ \hline 
  element of a set  	& \(\leftrightarrow\)	& sample point \\
  universal set		& \(\leftrightarrow\)	& sample space \\
  subset of a set	& \(\leftrightarrow\)	& event \\
 \end{tabular}
\end{frame}

\begin{frame}[t]{Example of a sample space}
 \begin{itemize}
  \item The experiment: You flip a quarter and a dime, and observe whether heads or tails on each coin. 
  \item What is the sample space?    \vfill
  \item What are the simple events?  \vfill
 \end{itemize}
\end{frame}

\begin{frame}[t]{The Empirical Probability}
  Given an event E in a sample space S, the \alert{empirical probability}, or \alert{relative frequency}, of E
  is given by 
  \[
    P(E)  =  \frac{n(E)}{n(S)}
  \]
  Think of this as ``probability is given by number of favorable outcomes divided by total number of outcomes.''
\end{frame}

\begin{frame}[t]{}
 \begin{itemize}
  \item The experiment: You flip a quarter and a dime, and observe whether heads or tails on each coin. 
  \item What is the probability you observe at least one heads?    \vfill
  \item What is the probability the dime shows a heads?\vfill
 \end{itemize}
\end{frame}



\begin{frame}[t]{Probability Spaces}
 A pair of fair six sided dice are rolled.  \\

 Determine an appropriate sample space.
\end{frame}

\begin{frame}[t]{Dice}
  \begin{center}
  \begin{tabular}{cccccc}
   \((1, 1)\)	&	\((2, 1)\)	&	\((3, 1)\)	&	\((4, 1)\)	&	\((5, 1)\)	&	\((6, 1)\)	\\
   \((1, 2)\)	&	\((2, 2)\)	&	\((3, 2)\)	&	\((4, 2)\)	&	\((5, 2)\)	&	\((6, 2)\)	\\
   \((1, 3)\)	&	\((2, 3)\)	&	\((3, 3)\)	&	\((4, 3)\)	&	\((5, 3)\)	&	\((6, 3)\)	\\
   \((1, 4)\)	&	\((2, 4)\)	&	\((3, 4)\)	&	\((4, 4)\)	&	\((5, 4)\)	&	\((6, 4)\)	\\
   \((1, 5)\)	&	\((2, 5)\)	&	\((3, 5)\)	&	\((4, 5)\)	&	\((5, 5)\)	&	\((6, 5)\)	\\
   \((1, 6)\)	&	\((2, 6)\)	&	\((3, 6)\)	&	\((4, 6)\)	&	\((5, 6)\)	&	\((6, 6)\)	\\
  \end{tabular}
  \end{center}
\begin{itemize}
  \item [a.)] Determine the event:  One die has an odd number and one die has either the number 1 or 2.
  \vfill
  \item [b.)] Determine the probability of the above event.
\end{itemize}
\end{frame}

\begin{frame}
  \begin{itemize}
   \item [c.)] 
    A pair of fair six sided dice is rolled.  
    Determine the probability that the sum of the values of the dice is exactly 4. 
    \vfill
   \item [d.)] 
    A pair of fair six sided dice is rolled.  
    Determine the probability that the sum of the value of the dice is at least 4. 
    \vfill
  \end{itemize}
\end{frame}

\begin{frame}[t]{Cards}
 \begin{itemize}
  \item One card is drawn at random from a standard deck of 52 cards.  
  \item Let \(A = \{\mbox{ card is a ``Diamond''}\}\), \(B = \{\mbox{ card is a ``5''}\}\), \(C = \{\mbox{card is a ``King''}\}\).
  \item \(P(A) = \)
  \item \(P(B) = \)
  \item \(P(C) = \)
  \item \(P(B \cup C) = \)
  \item \(P(A \cap B) = \)
  \item \(P(A \cup B) = \)
  \item \(P(B \cap C) = \)
 \end{itemize}
\end{frame}

\begin{frame}[t]{Mutually Exclusive Events}
 \begin{itemize}
  \item Two events are \alert{Mutually Exclusive} if they cannot occur simultaneously
  \item In other words, \(E_1 \cap E_2 = \emptyset\)
 \end{itemize}
\end{frame}


%\begin{frame}[t]{Probability Measure}
% \begin{itemize}
%  \item Consider an experiment with sample space \(S = \{s_1, s_2, \ldots, s_n\}\). \vfill
%  \item A probability function, P, assigns a to each event a number between 0 and 1 according to the rules: \vfill
%  \begin{itemize}
%   \item \( 0 \leq P(E) \leq 1\) for each event E. \vfill
%   \item \( P(s_1) + P(s_2) + \ldots + P(s_n) = 1\) \vfill
%   \item If events \(E_1\) and \(E_2\) are mutually exclusive then 
%    \[
%    P(E_1 \cup E_2) = P(E_1) + P(E_2)
%    \] \vfill
%  \end{itemize}
% \end{itemize}
%\end{frame}

\begin{frame}[t]{Empirical Distribution: Example}
200 students were surveyed regarding the number of credit hours they are taking in the current semester.  
Here are their responses:
\begin{tiny}
\begin{center}
  \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline
   Credit hours    &  12 or fewer & 13 hours & 14 hours & 15 hours & 16 hours & 17 hours & 18 or more \\ \hline
   Number students &      46      & 51 & 29 & 55 & 12 &  6 &  1         \\ \hline
\end{tabular}
\end{center}
\end{tiny}
A student is selected at random from this group.  What is the probability that they are taking at least 15 credit hours?
\end{frame}


\begin{frame}[t]{7.2: Another Empirical Distribution}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline
Age           &   85 &   86 &   87 &   88 &   89 &   90 &   91  \\ \hline
Number deaths & 3590 & 3711 & 3796 & 3838 & 3828 & 3757 & 3624  \\ \hline
\end{tabular}
The above is an excerpt from the Social Security Administrations Mortality Table.
The table follows a cohort of 100,000 people and records the age of death of these individuals.
\begin{itemize}
 \item Determine the probability a randomly chosen individual dies at age 90.
 \item Determine the probability a randomly chosen individual dies between age 87 and 90.
\end{itemize}
\end{frame}

\end{document}

