\documentclass[12pt]{article}
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\usepackage[homework]{quizmachine}

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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Homework 8}
\date{due May 2, 2013, before 12:00}

\quizleft{ESM 2B}
\quizcenter{Quiz 8}
\quizright{May 8, 2013}

\hwpreamble{\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.}

\defproblem{
\begin{itemize}
\item[(E)] Determine whether the following relationships are valid.
For those that are, prove the relationship using de Morgan's law.
\begin{enumerate}
\item $\overline{(\overline X \cup Y)} = X \cap \overline Y$
\item $(X \cup Y) \cap Z = (X \cup Z) \cap Y$
\item $X\cup \overline{(Y \cap Z)}=(X \cup \overline Y) \cup \overline Z$
\end{enumerate}

\item[(A)] Given that events $X,Y$, and $Z$ satisfy
\[
  (X \cap Y) \cup (Z \cap X) \cup \overline{(\overline X \cup
  \overline Y)}
  = \overline{(Z \cup \overline Y)} \cup
    \bigl[
    \bigl(
      \overline{(\overline Z \cup \overline X)} \cup
      (\overline X \cap Z)
    \bigr) \cap Y
    \bigr] \,,
\]
prove that $X\supset Y$ and that $Z \cap X \subset Y$.
\end{itemize}}


\defproblem{
Let $A$ and $B$ be two mutually exclusive 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
$P(B-A)$.}

\defproblem{ 
Gamblers $A$ and $B$ each roll a fair six-faced die, and $B$
wins if his score is strictly greater than $A$'s.  Show that the odds
are $7$ to $5$ in $A$'s favor.}

\defproblem{Let $A$ and $B$ two events which are independent and
mutually exclusive.  Show that at least one of them is an
impossibility.} 

\defproblem{
You witness a night-time hit-and-run accident involving a taxi
in a city in which 9 out of 10 taxis are green, and all other taxis
are blue.  Discrimination between the two colors is 80\% reliable
under the dim lighting conditions at the scene.  You are sure that the
taxi was blue.  What is the most likely color of the taxi, and what is
the probability that your impression is correct?}

\defproblem{ 
Bayesylvania is awash with natives not all of whom can be trusted to
tell the truth.  A traveler finds himself at a T-junction in an area
populated by the Asciis and Bisciis in the ratio 11 to 5.  A Biscii
always lie but an Ascii tells the truth three quarters of the time,
giving independent answers to all questions, even to immediately
repeated ones.

\begin{itemize} 
\item[(E)] The traveler asks one particular native whether he should
go to the left or to the right to reach the local village.  He is told
``left''.  Should he take this advice, and, if he does, what are his
chances of reaching the village?

\item[(A)] The traveler asks one particular native three times whether
he should go to the left or to the right to reach the local village.
Each time he receives the same answer ``left''.  What should the
traveler do now?  Have his chances of finding the village been altered
by asking the question three times?
\end{itemize}}

\quizzes{100}

\end{document}
