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\section*{CS 70 homework 10 solutions}
Your full name: PUT YOUR NAME HERE
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Your login name: PUT YOUR LOGIN NAME HERE
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Homework 10
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Your section number: PUT YOUR SECTION NUMBER HERE
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Your list of partners: LIST YOUR PARTNERS HERE
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\begin{enumerate}
\item
Here is some on-time arrival data for two airlines, A and~B, into
the airports of Los Angeles and Chicago. (Predictably, both airlines
perform better in LA, which is subject to less flight
congestion and less bad weather.)
\begin{center}
\begin{tabular}{l|cc|cc|}
&\multicolumn{2}{c}{Airline A}&\multicolumn{2}{c}{Airline B}\\
&\# flights&\# on time&\#flights&\# on time\\ \hline
Los Angeles&600&534&200&188\\
Chicago&250&176&900&685\\ \hline
\end{tabular}
\end{center}
\begin{enumerate}
\item Which of the two airlines has a better chance of
arriving on time into Los Angeles? What about Chicago?
YOUR ANSWER GOES HERE.
\item Which of the two airlines has a better chance of
arriving on time overall?
YOUR ANSWER GOES HERE.
\item Do the results of parts~(a) and (b) surprise you?
Explain the apparent paradox, and
interpret it in terms of conditional probabilities.
YOUR ANSWER GOES HERE.
\end{enumerate}
\item
Let $\Omega$ be a sample space, and let $A,B \subseteq \Omega$ be two
\emph{independent} events.
Let $\overline{A} = \Omega-A$ and $\overline{B} = \Omega-B$
(sometimes written $\neg A$ and $\neg B$)
denote the complementary events.
For the purposes of this question, you may use the following
definition of independence: Two events $A,B$ are
\emph{independent} if $\Pr[A \cap B] = \Pr[A] \Pr[B]$.
\begin{enumerate}
\item Prove or disprove:
$\overline{A}$ and $\overline{B}$ are necessarily independent.
YOUR ANSWER GOES HERE.
\item Prove or disprove:
$A$ and $\overline{B}$ are necessarily independent.
YOUR ANSWER GOES HERE.
\item Prove or disprove:
$A$ and $\overline{A}$ are necessarily independent.
YOUR ANSWER GOES HERE.
\item Prove or disprove: It is possible that $A=B$.
YOUR ANSWER GOES HERE.
\end{enumerate}
\item
When $\Pr[A\vert B] > \Pr[A]$, then $A$ and $B$ may be viewed
intuitively as being positively correlated.
One might wonder whether ``being positively correlated''
is a symmetric relation.
Prove or disprove: If $\Pr[A\vert B] > \Pr[A]$ holds,
then $\Pr[B\vert A] > \Pr[B]$ must necessarily hold, too.
(You may assume that both $\Pr[A\vert B]$ and $\Pr[B\vert A]$
are well-defined, i.e., neither $\Pr[A]$ nor $\Pr[B]$ are zero.)
YOUR ANSWER GOES HERE.
\item
[\ldots]
Fill in the table below to show player B's best choice of triplet
for each possible choice that player A makes, and the probability
of player B winning with a best choice.
(We suggest you determine this information with a computer program,
which you should submit with your solutions.)
\begin{tabular}{|c|c|c|} \hline
Player A's choice&Player B's best choice&Player B's probability of winning\\ \hline
HHH&FILL ME IN &FILL ME IN \\ \hline
HHT&FILL ME IN &etc. \\ \hline
HTH& & \\ \hline
HTT& & \\ \hline
THH& & \\ \hline
THT& & \\ \hline
TTH& & \\ \hline
TTT& & \\ \hline
\end{tabular}
Then explain why the odds for one player winning are so lopsided.
YOUR ANSWER GOES HERE.
\item
It's a hot summer day in the Central Valley. [\ldots]
What are the survival probabilities for each of Alice, Bob, and Carlos?
(Make clear how you got your answer.)
YOUR ANSWER GOES HERE.
\end{enumerate}
\end{document}