% 302f17Assignment3.tex             Conditional probability and independence
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{\Large \textbf{\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f18}{STA 256f18} 
Assignment Three}}\footnote{Copyright information is at the end of the last page.}
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\noindent
Please read Sections 1.5-1.7 (pages 16-26) in the text, and look over your lecture notes. These homework problems are not to be handed in. They are preparation for Term Test 1 and the final exam.  All textbook problems are from Chapter One.

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%%%%%%%%%%%%%%%%%%%% Conditional probability and independence %%%%%%%%%%%%%%%%%%%%

\item Do Problem 45 in the text.
\item Do Problem 46 in the text.
\item Do Problem 47 in the text.
\item Do Problem 53 in the text.

\item I die is a cube with 1, 2, 3, 4, 5 or 6 dots on each face. Roll two fair dice. 
    \begin{enumerate}
        \item What is the probability that the two numbers are different?
        \item What is the probability that the sum is even?
    \end{enumerate}

\item Do Problem 58 in the text. This is one version of a classic problem, and the reason it's a classic is that it's so easy to get mixed up. Use the definition of conditional probability. What is the probability of Drew given that the teacher says ``Chris?".

\item Do Problem 59 in the text.
\item Do Problem 60 in the text.
\item Do Problem 62 in the text.
\item Do Problem 63 in the text.
\item Do Problem 64 in the text.
\item Do Problem 65 in the text.
\item Do Problem 68 in the text.
\item Do Problem 69 in the text.
\item Do Problem 70 in the text.
\item Do Problem 72 in the text.
\item Do Problem 74 in the text.
\item Do Problem 77 in the text.

\item Roll a single fair die repeatedly.
    \begin{enumerate}
        \item What is the probability that the first 6 appears on the 4th roll?
        \item What is the probability that a 6 eventually occurs -- that is, on roll 1 or 2 or \ldots? Show your work.
        \item What is the probability that the first 6 occurs on an odd numbered roll?
    \end{enumerate}

\item A jar contains 10 red balls and 20 blue balls. If you sample 5 balls randomly \emph{with} replacement, what is the probability of
    \begin{enumerate} % Straight from lecture
        \item All blue?
        \item At least one red?
        \item Two red and three blue?
        \item Obtaining $j$ red balls, $j = 0, \ldots, 5$? Give a single formula. Don't simplify.
    \end{enumerate}

\item Let $\Omega = \cup_{k=1}^\infty B_k$, disjoint, with $P(B_k)>0$ for all $k$.
    \begin{enumerate}
        \item Using the formula sheet and the tabular format illustrated in lecture, prove the Law of Total Probability: $P(A) = \sum_{k=1}^\infty P(A|B_k)P(B_k)$.
        \item Prove the following version of Bayes' Theorem: $ P(B_j|A) = \frac{P(A|B_j)P(B_j)}{\sum_{k=1}^\infty P(A|B_k)P(B_k)}$. You may use anything from the formula sheet except Bayes' theorem itself.
    \end{enumerate}

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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: 
     
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\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f18}{STA 256f18}