% Sample Question document for STA256
\documentclass[12pt]{article} 
%\usepackage{amsbsy} % for \boldsymbol and \pmb 
%\usepackage{graphicx} % To include pdf files!
\usepackage{amsmath}
\usepackage{amsbsy}
\usepackage{amsfonts}
\usepackage[colorlinks=true, pdfstartview=FitV, linkcolor=blue, citecolor=blue, urlcolor=blue]{hyperref} % For links
\usepackage{fullpage}
%\pagestyle{empty} % No page numbers


\begin{document}
%\enlargethispage*{1000 pt} 

\begin{center}   
{\Large \textbf{Sample Questions: Foundations of Probability}}%\footnote{}
\vspace{1 mm}

STA256 Fall 2018. Copyright information is at the end of the last page.
\end{center}

\vspace{5mm}
\begin{enumerate} 

\item Prove $P(A^c) = 1-P(A)$. Use the axioms of probability and the tabular format illustrated in lecture.
\pagebreak

\item  Prove $P(\emptyset)=0$. Use the axioms of probability and the tabular format illustrated in lecture.
\pagebreak

\item  Prove that if $A \subseteq B$ then $P(A) \leq P(B)$. Use the axioms of probability and the tabular format illustrated in lecture.
\pagebreak

\item  Prove the Addition Law: $P(A \cup B) = P(A)+P(B)-P(A\cap B)$. Use the axioms of probability and the tabular format illustrated in lecture.
\pagebreak

\item If 23 out of 25 are employed, what is the probability of randomly choosing an unemployed person?  The answer is a number. Circle your answer.
\pagebreak

\item If you roll two fair dice, what is the probability of getting a sum greater than 2? The answer is a number. Circle your answer.
\pagebreak

\item If you roll two fair dice, what is the probability of getting two different numbers? Your answer is a number. Circle your answer.
\pagebreak

\item $P(A)=0.4$, $P(B)=0.5$ and $P(A\cap B)=0.3$. What is $P(A\cup B)$? The answer is a number. Circle your answer. 
\pagebreak

\item Of the cars in a used car lot, 50\% have engine trouble and 50\% have transmission trouble. If 25\% have both problems and you buy a car at random, what is the probability that both the engine and transmission are okay? The answer is a number. Circle your answer. 
\pagebreak

\item Of the prisoners in a jail, 75\% are convited murderers and 50\% have been convicted of both murder and armed robbery. Twenty percent are in jail for offences other than murder or armed robbery. If you pick a prisoner at random, what is the probability that she is an armed robber?


\end{enumerate}

\vspace{150mm}

\noindent
\begin{center}\begin{tabular}{l}
\hspace{6in} \\ \hline
\end{tabular}\end{center}
This assignment was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Mathematical and Computational 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: 
     
\begin{center}
\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f18} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f18}}
\end{center}

\end{document}