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\textbf{\Large MAT 3701:  Assignment 2 \hfill My Name}
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%\emph{Explain all answers completely in essay form!  (Neat hand-done diagrams may be inserted or attached, although you can also make these with various software options, or scan them to PDF files, and easily insert them into your \emph{Tex} document.)}



\section{Exercises from the text}

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\item %This is exercise 2.1, which has three parts to be answered below.

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\item %Fill in your solution to part (a) here.

\item %Fill in your solution to part (b) here.  

\item %Fill in your solution to part (c) here.

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\item %Fill in your solution to part (a) here.

\item %Fill in your solution to part (b) here.

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\item %Fill in your solution to Exercise 3 here. Etc.

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\section{Additional Exercises}

\emph{Explain all answers completely in essay form!}

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%\noindent \textbf{N.B. (This common abbreviation stands for \emph{nota bene}, which is Latin for \emph{note well};  in other words, this remark is important!)}  \emph{There is insufficient information given to answer one of the questions below.  For this question, please briefly explain why the information given is insufficient.)}

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\item Given integers $n$ and $k$, suppose we wish to choose $n$ integers whose sum is less than or equal to $k$,  that is, integers $i_1, i_2, i_3, \ldots, i_n$ such that $\sum_{j=1}^n i_j \leq k$.  Show that the number of ways this can be done is $\binom{n+k}{n}$.  Note that the integers $\{i_j\}_{j=1}^n$ do not have to be distinct.  (Big hint:  Imagine a row of $n+k$ tokens (sticks, pennies, or whatever).  Choose $n$ of these to remove, and let the numbers $i_1, i_2, i_3, \ldots, i_n$ be the number of tokens in the first $n$ groups separated by the those that have been removed.  There may be a non-empty $(n+1)^{{st}}$ group of leftovers, in which case the sum is strictly less than $k$.)

\item Given integers $n$ and $k$, suppose we wish to choose $n$ integers whose sum is \emph{exactly} equal to $k$,  that is, integers $i_1, i_2, i_3, \ldots, i_n$ such that $\sum_{j=1}^n i_j = k$.  Show that the number of ways this can be done is $\binom{n+k-1}{n-1}$.  (Hint:  In this case, we don't allow any leftovers.)  In addition, explain how this calculation is related to occupancy problems.)

\item Given events $E_1, E_2, E_3, \ldots, E_n$ from a sample space $S$ with probability measure $\rho$, let $\displaystyle S_k = \sum_{1 \leq i_1 < i_2 < i_3 < \cdots < i_k \leq 1}E_{i_1} \cap E_{i_2} \cap E_{i_3} \cdots \cap E_{i_k}$.  Prove that $\displaystyle \rho(E_1 \cup E_2 \cup E_3 \cup \cdots \cup E_n) = \sum_{k = 1}^n S_k$.

\item If a (fair) coin is tossed $n$ times, what is the probability that a particular given sequence of heads and tails will be tossed?  (Hint:  Each possible sequence is equally likely.  How many are there?)

\item If a coin is tossed three times, what is the probability that it comes up heads two out of the three times?

\item  Suppose a coin is tossed $n$ times.  What is the probability that it comes up heads at least once?

\item An urn contains 5 red balls, 2 black balls, and 3 white balls.  (Probability theorists are inordinately fond of urns, the idea presumably being that one cannot see into them.)  Three balls are picked at random.

\begin{enumerate}

\item What is the probability that one ball of each color is picked?

\item What is the probability that all three red balls are picked?

\item What is the probability that no red balls are picked?

\item What is the probability that two black balls are among those picked?

\item What is the probability that two black balls and one white ball are picked?

\item If you know that one of the balls is black, what is the probability that the other two are red?

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\item Suppose for the urn above that first three balls are picked and placed in a bowl, then another three balls are picked and placed in a second bowl, and finally the remaining three balls are picked and placed in a third bowl.  

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\item What is the probability that each bowl contains a white ball?

\item What is the probability that each bowl contains a red ball?

\item What is the probability that the there are two red balls in the first bowl, three in the second, and none in the third?

\item What is the probability that both black balls are in the same bowl?

\end{enumerate}

\item Suppose a spinner is constructed that is equally likely to stop at any point of the compass when spun.  What is the probability that it will land between $0^\circ$ (North) and $22^\circ$?

\item  If we know the spinner landed between $0^\circ$ (North) and $22^\circ$ but don't know exactly where it landed, what is the probability that it lies between $5^\circ$ (North) and $10^\circ$?


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