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{\Large \textbf{STA 256f19 Assignment Two}}\footnote{Copyright information is at the end of the last page.}
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\noindent
Please read Sections 1.2, 1.3 and 1.4 in the text (pages 4-20) and review lecture sets entitled \emph{Sets}, \emph{Foundations of Probability} and \emph{Counting}. These homework problems are not to be handed in. They are preparation for Term Test 1 and the final exam. Use the formula sheet. % Some of the questions are too easy to be on the test or exam, but they are good preparation for the real problems. 

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\begin{enumerate} 

%%%%%%%%%%%%%%%%%%%% Sets %%%%%%%%%%%%%%%%%%%%

\item Do Exercises 1.2.6 and 1.2.7 in the text. For 1.2.6, just give regions $a$, $b$ and $e$. The answers are $a = A \cap B^c \cap C^c$, $b = A \cap B \cap C^c$, and $e = A \cap B \cap C$.

% Sample Question
\item Make Venn diagrams to illustrate the distributive laws:
    \begin{enumerate}
        \item $(A \cup B) \cap C = (A \cap C) \cup (B \cap C)$
        \item $(A \cap B) \cup C = (A \cup C) \cap (B \cup C)$
    \end{enumerate}

% Sample Question
\item Make Venn diagrams to illustrate the De Morgan laws:
    \begin{enumerate}
        \item $(A \cap B)^c = A^c \cup B^c$
        \item $(A \cup B)^c = A^c \cap B^c$
    \end{enumerate}

% Sample Question
\item Make a Venn diagram showing that if $A$ and $B$ are disjoint, then $A \cap C$ and $B \cap C$ are also disjoint.

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%%%%%%%%%%%%%%%%%%%% Foundations %%%%%%%%%%%%%%%%%%%%

% Sample Question
\item Prove Property 5: $P(A^c) = 1-P(A)$. Use Properties 1-4 of probability and the tabular format illustrated in lecture.

% Sample Question
\item  Prove Property 6:  If $A \subseteq B$ then $P(A) \leq P(B)$. Use Properties 1-4 of probability and the tabular format illustrated in lecture.

% Sample Question
\item  Prove Property 7 (the inclusion-exclusion principle): $P(A \cup B) = P(A)+P(B)-P(A\cap B)$. Use Properties 1-4 of probability and the tabular format illustrated in lecture.

\item Do Exercises 1.2.1, 1.2.3 and 1.2.4 in the text. The answer to 1.2.4 is No. Consider $P(\{ 2,3\})$.

\item Do problems 1.2.13, 1.2.14 and 1.2.15. The answers are No, No, Yes. Here is some background information you may not have seen yet. An infinite set is said to be \emph{countable} if it can be placed in a one-to-one correspondence with the set of natural numbers $1, 2, \ldots$. By Cantor's famous diagonalization proof, the real numbers between zero and one are not countable. Thus, adding up their probabilities is not possible.

% Sample Question
\item  Let $A_1, A_2, \ldots$ form a partition of the sample space $S$, meaning that $A_1, A_2, \ldots$ are disjoint and $S = \cup_{k=1}^\infty A_k$. Let $B$ be any event. Show that $P(B) = \sum_{k=1}^\infty P(A_k \cap B)$. Use Properties 1-4 of probability and the tabular format illustrated in lecture.

% Sample Question
\item  Let $A_1, A_2, \ldots$ be a collection of events, not necessarily disjoint. Show that
    $P\left( \cup_{k=1}^\infty A_k\right) \leq \sum_{k=1}^\infty P(A_k)$.  Use the Properties 1-7 of probability and the tabular format illustrated in lecture.
    
\item Do Exercises 1.3.1, 1.3.3 and 1.3.5. For Exercise 1.3.5, what is the sample space? Assume all outcomes are equally likely. 

\item Do Problem 1.3.9 in the text. It may be helpful to make a Venn diagram. The answer is that $P(\{2\})$ could be as small as zero, and as large as 0.2. 

\item  Let $A_1 \subseteq A_2 \subseteq A_3 \subseteq \ldots$ and let $A = \cup_{k=1}^\infty A_k$. Show that $\lim_{k \rightarrow \infty} P(A_k) = P(A)$.  Use Properties 1-4 of probability and the tabular format illustrated in lecture. % Straight from Lecture

\item  Let $A_1 \supseteq A_2 \supseteq A_3 \supseteq \ldots$ and let $A = \cap_{k=1}^\infty A_k$. Show that $\lim_{k \rightarrow \infty} P(A_k) = P(A)$.  Use Properties 1-4 of probability and the tabular format illustrated in lecture. % Straight from Lecture



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%%%%%%%%%%%%%%%%%%%% Counting %%%%%%%%%%%%%%%%%%%%

\item If eight children are standing in line,
    \begin{enumerate}
        \item In how many orders can they stand?
        \item In how many orders can they stand if two friends insist on being together?
        \item Suppose there are four boys and four girls. In how many orders can they stand if the boys stay together and the girls stay together?
        \item If the children line up completely at random, what is the probability that the four boys are together and the four girls are together?
    \end{enumerate}

\item The four players in a bridge game are each dealt 13 cards from an ordinary 52-card deck. How many ways are there to do this? The answer has 29 digits, so stop simplifying after you arrive at a multinomial coefficient. % This is expanded from problem 14 in Rice chapter one.

\item In a game of poker, four players are each dealt 5 cards from a 52-card deck. How
many ways are there to deal the cards? % Straight from lecture. Multinomial.

\item A box of 25 Valentine's Day chocolates has 10 that are cream filled and 15 that are not cream filled. If you eat 7 chocolates at random, what is the probability that you get exactly 2 cream filled? Just write down the answer. There is no need to simplify. This question was on the 2018 final exam.

\item This question is taken from \emph{Mathematical statistics and data analysis}, by Rice. A drawer of socks contains seven black socks, eight blue socks, and nine green
socks. Two socks are chosen in the dark.
    \begin{enumerate}
        \item What is the probability that they match?
        \item What is the probability that a black pair is chosen?
    \end{enumerate}

\item Do Exercise 1.4.1 in the text. % 8 dice

\item Do Exercise 1.4.4 in the text. % Poker

\item Do Exercise 1.4.5 in the text. % Simple bridge

\item Do Exercise 1.4.7 in the text. % Deal 10 cards until 1st Jack.

\item Do Exercise 1.4.11 in the text. The sampling is without replacement.

\item A jar contains 10 red balls and 20 blue balls. If you sample 5 balls randomly without replacement, what is the probability of
    \begin{enumerate} % Straight from lecture
        \item All blue?
        \item Two red and three blue?
        \item At least one red?
        \item Obtaining $j$ red balls, $j = 0, \ldots, 5$? Give a single formula. Don't simplify.
    \end{enumerate}

\item Do Challenge problem 1.4.21 in the text. The answers are not in the back of the book. They are
    \begin{enumerate} 
        \item[(a)] $\frac{365 \cdot 364}{365^2}$
        \item[(b)] $\frac{365}{365^C}$
        \item[(c)] $1-  \frac{365 \cdot 364 \cdots (365-C+1) }{365^C}  = 1-  \frac{_{365} P_C}{365^C}  = 1-  \frac{365!}{(365-C)!365^C}$
        \item[(d)] $C=23$. I had to write a program to find this, so you could not be expected to do it on a test or the final exam. 
    \end{enumerate}

\end{enumerate}

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\noindent
Here are some more answers that are not in the back of the book.
        \begin{itemize}
            \item[16.] 
                \begin{itemize}
                    \item[(a)] $8!$
                    \item[(b)] $7! \cdot 2$
                    \item[(c)] $4! 4! \cdot 2$
                    \item[(d)] $\frac{4! 4! \cdot 2}{8!} \approx 0.02857$
                \end{itemize}
            \item[17.] $\binom{52}{13~13~13~13}$
            \item[18.] $\binom{52}{5~5~5~5~32}$ 

            \item[19.] $\frac{\binom{10}{2}\binom{15}{5}}{\binom{25}{7}}$ 
            \item[20.] 
                \begin{itemize}
                    \item[(a)] $\frac{\binom{7}{2} + \binom{8}{2} + \binom{9}{2}} {\binom{24}{2}}$
                    \item[(b)] $\frac{\binom{7}{2}} {\binom{24}{2}}$
                \end{itemize}
        \end{itemize}



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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. Except for the questions that are borrowed from taken from \emph{Mathematical statistics and data analysis}, by Rice, 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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% These might be independence

\item \label{birthday} Suppose that a room contains $n$ people. What is the probability that at least two of them have a common birthday? Assume a year of 365 days, and that the chances of being born on all days are equal. % Example E in Rice, Page 10. 

\item Under the assumptions of Problem~\ref{birthday}, what values of $n$ is needed so that the probability is at least 50\%?


