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\begin{document}

\centerline{Homework 10, Morally due Tue Apr 30, 3:30PM}

\centerline{\bf THIS HW IS THREE PAGES!!!!!!!!!!}

\newif{\ifshowsoln}
\showsolntrue % comment out to NOT show solution inside \ifshowsoln and \fi blocks.

\begin{enumerate}

\item
(0 points but if you don't show up to the final I will assume you got this
problem wrong and you will get 0 points for this entire HW)
WHEN IS THE FINAL?
WHERE IS THE FINAL?

\item
(30 points)
\begin{enumerate}
\item
(15 points)
Josh rearranges the letters in the sequence  {\it machinery} randomly.
What is the probability that the new sequences is {\it machinery}
\item
(15 points)
Bill makes lunch for her darling.
There is a sandwich- either PBJ, Turkey, Tomato, Egg salad, or Tuna fish,
a fruit- either apple or blueberries or blackberries or a banana,
and a snack- either pretzels, potato chips or applesauce.
Suppose Bill selects a lunch to prepare uniformly at random out of all the possibilities.
 What is the probability that Bill's darling gets a lunch that DOES NOT
have both an apple and applesauce.
\end{enumerate}

\centerline{\bf GO TO NEXT PAGE}

\newpage

\item
(40 points)
I have two coins.

One of them is FAIR

One of them is BIASED: Prob(H)=$\frac{7}{12}$, Prob(T)=$\frac{5}{12}$.

One is chosen at random (prob 1/2 for each). That coin is tossed 20 times.

Do the following TWENTY ONE problems and put them in a table. 
For the first one show us your work (you can use a calculator or your program for the
arithmetic), but the rest just have the answers in the table.

You will want to write a computer
program for them.  Note when the prob of biased goes from $>\frac{1}{2}$ to $<\frac{1}{2}$.

\begin{itemize}
\item
The result is $HHHHHHHHHH$ (so 20 H's and 0 T).  What is the prob that the coin is biased?
\item
The result is $HHHHHHHHHT$ (so 19 H's and 1 T).  What is the prob that the coin is biased?
\item
The result is $HHHHHHHHTT$ (so 18 H's and 2 T).  What is the prob that the coin is biased?
\item
$\vdots$
\item
The result is $TTTTTTTTTT$ (so 0 H's and 20 T).  What is the prob that the coin is biased?
\end{itemize}


All numbers should be to six places, so for example

$$(7/12)^{20} \sim 0.000021$$

\centerline{\bf GO TO NEXT PAGE}

\newpage

\item
(30 points)
I have two 10-sided die.

One of them is FAIR

One of them is BIASED: Prob(1)=Prob(10)=$\frac{1}{2}$ and Prob(2)=$\cdots$=Prob(9)=0.

\begin{enumerate}
\item
I roll the fair die.  What is the expected value? What is the variance?

\item
I roll the biased die.  What is the expected value? What is the variance?

\item
I roll both and add the values.  What is the expected value? What is the variance?
\end{enumerate}


\end{enumerate}

\end{document}