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\begin{document}

\centerline{Homework 9, Morally due Tue Apr 23, 3:30PM}

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

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

\begin{enumerate}

\item
(40 points)
Throughout this problem Bill has a 2-sided dice with numbers 1,2 and a 3-sided die with numbers 1,2,3.
\begin{enumerate}
\item
(15 points)
Assume both dice are fair. 
Bill throws both of them. For $2\le i\le 5$ give the prob that the sum is $i$.
\item
(20 points)
Let $0\le p\le \frac{1}{2}$.
Assume the 2-sided dice is fair but the 3-sided dice has

Prob of 1 = $p$

Prob of 2 = $1-2p$

Prob of 3 = $p$

Bill throws both of them. For $2\le i\le 5$ give the prob that the sum is $i$.

\item
(5 points)
Let $p$ be as in the last part.
Is there a value of $p$ such that all of the sums $2,3,4,5$ come up with the same probability.

\item
(0 points but thing about it) Can you load two 6-sided dice to get fair sums?
\end{enumerate}


\centerline{\bf GO TO NEXT PAGE}

\newpage

\item
(60 points)
On the planet Vorlon they play a game that is similar to what we call Poker
but with a different deck of cards.

Every card has a rank from $\{1,2,\ldots,7\}$.

Every card has a suite from $\{R,B\}$.

Every player gets 3 cards.

In most of the questions we will ask for the prob of a certain
type of hand. Give the answer to 4 places since the last question is to rank them.

\begin{enumerate}
\item
What is prob of a straight that is NOT a flush (e.g., $3R$, $4R$, $6B$)
We DO allow wrap-around, so 7-1-2 counts.
\item
What is prob of a  flush that is NOT a straight (e.g., $2R$, $4R$, $9R$)
\item
What is prob of a  straight flush (e.g., $3R$, $4R$, $6R$)
We DO allow wrap-around, so 7-1-2 counts.
\item
What is prob of a pair (e.g., $3R$, $4B$, $7R$). Note that a pair cannot be a
straight of a flush.
\item
What is prob of getting NOTHING- a hand that is neither a straight, nor a flush, nor does it contain 2 of a kind.
(e.g., $3R$, $5R$, $6B$)
\item
Rank the types of hands from most likely to least likely.
\end{enumerate}



\end{enumerate}

\end{document}