% 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: Independence}}

STA256 Fall 2018. Copyright information is at the end of the last page.
\vspace{1 mm}
\end{center}

\vspace{5mm}
\begin{enumerate} 

\item A jar contains 5 red balls and 15 black balls. Draw 2 balls randomly with replacement. 
    \begin{enumerate}
        \item What is the probability that the first ball is red and the second is black? The answer is a number. \vspace{85mm}
        \item What is the probability of one red and one black in any order? The answer is a number. 
    \end{enumerate}

\pagebreak

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item Roll a fair die $n$ times. 
    \begin{enumerate}
        \item What is the probability of observing at least one 4?  \vspace{85mm}
        \item How many times must you roll the die for the probability of at least one 4 to be 0.90 or more? The answer is a number. 
    \end{enumerate}

\pagebreak

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item A biased coin has $P(\mbox{Head}) = p$. Toss it three times.
    \begin{enumerate}
        \item List the elements of the sample space, along with their probabilities.  \vspace{70mm}
        \item What is $P(\mbox{Two Heads})$?  \vspace{20mm}
    \end{enumerate}

\item It is clear from the last problem that the probability of a string with $k$ heads is the same, regardless of their placement. Suppose we toss the biased coin $n$ times. What is the probability of $k$ heads (for $k = 0, \ldots, n$)? 

\pagebreak

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item Again, a biased coin has $P(\mbox{Head}) = p$. Toss it until the first head occurs, and then stop. 
    \begin{enumerate}
        \item What is the probability that the first head appears on the fifth toss? \vspace{30mm}
        \item What is the probability that a head eventually occurs (on toss 1 or 2 or \ldots)? \pagebreak
        \item  What is the probability that the first head occurs on an even numbered toss \linebreak
        (toss 2 or 4 or \ldots)?
    \end{enumerate}

\end{enumerate}

\vspace{160mm}

\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}

% The answer is a number. Circle your answer.
