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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Homework 9}
\date{due May 5, 2004, before 12:00}
\maketitle

\noindent
Normal questions and advanced questions (A) are worth $5$ points; easy
questions (E) are worth $4$ points.  Complete either the easy, or the
advanced version, not both.

\begin{enumerate}

\item Let $A$ and $B$ be two \emph{statistically independent} events.
Suppose $P(A)=\tfrac13$ and $P(B)=\tfrac14$.  Compute the
probabilities $P(A|B)$, $P(B|A)$, $P(A\cup B)$, $P(A\cap B)$,
$P(A-B)$, and $P(B-A)$.

\item
A boy is selected at random from among the children belonging to
families with $n$ children.
\begin{itemize} 
\item[(E)] What is the probability that the boy has $k-1$ brothers?

\item[(A)] It is known that the boy has at least two sisters.  Show
that the probability that he has $k-1$ brothers is
\[
  \frac{(n-1)!}{(2^{n-1}-n) \, (k-1)! \, (n-k)!}
\]
when $1 \leq k \leq n-2$, and zero for other values of $k$.

Hint: Use part (E) and Bayes' rule.
\end{itemize}

\item Gamblers $A$ and $B$ each have two unbiased four-sided dice, the
four faces being numbered $1$, $2$, $3$, $4$.  Without looking, $B$
tries to guess the sum $x$ of the numbers on the bottom faces of $A$'s
two dice after they have been thrown onto a table.  If the guess is
correct, $B$ receives $x^2$ Euros, but if not he loses $x$ Euros.

\begin{itemize}
\item[(E)] Show that, when guessing the sum of $x$, $B$'s expected
gain $G$ per throw of $A$'s dice is
\[
  E[G_x] = p_x \, (x^2+x) - x \,,
\]
where $p_x$ is the probability that the sum of the bottom faces is
$x$.

\item[(A)] Compute the expected gain of $B$ if he always guesses the
sum of $A$'s bottom faces from the previous round.
\end{itemize}

\item In how many ways can $8$ people be placed around a table if
there are three who insist on sitting together?

\item Prove the following identities:

\begin{itemize} 
\item[(E)] $\displaystyle \sideset{^n}{_k}{\operatorname{C}} \,
\sideset{^k}{_\ell}{\operatorname{C}} =
\sideset{^n}{_\ell}{\operatorname{C}} \,
\sideset{^{n-l}}{_{k-\ell}}{\operatorname{C}}$

\item[(A)] $\displaystyle \sum_{i=0}^k
\sideset{^m}{_i}{\operatorname{C}} \,
\sideset{^n}{_{k-i}}{\operatorname{C}} =
\sideset{^{m+n}}{_k}{\operatorname{C}}$
\end{itemize}

\item A royal family has children until it has a boy or until it has
three children, whichever comes first. Assume that each child is a boy
with probability $\tfrac12$. Find the expected number of boys in this
family and the expected number of girls.


\end{enumerate}
\end{document}
