% 35 copies

\documentclass[12pt,twoside]{article}
\input{macros-course}
\begin{document}
\handout{hw2}{Optional Homework 2}{July 29, 2003}

\begin{problems}

\problem Two coins are given. One is fair (the probability of heads is
$1/2$), and the other is biased with a $2/3$ probability of heads.
One of the coins is tossed once, resulting in heads.  The other is
tossed three times, resulting in two heads.  Which coin is more likely
to be the biased one?

\problem Anu and Bing play a game.  Anu picks a $3$-pattern from the
$8$ possible $3$-patterns of coin flips:
\[
\begin{array}{cccc}
HHH & HHT & HTH & HTT\\
THH & THT & TTH & TTT
\end{array}
\]
Then, Bing picks a different $3$-pattern.  They now flip a fair coin
as many times as it takes for one of their $3$-patterns to show up as
consecutive coin tosses, the winner being the person whose pattern
shows up first.  For example, if Anu picks $HHT$ and Bing picks $TTH$,
then Anu would win if the sequence of coin flips were
$H\,T\,H\,T\,H\,H\,\underline{H}\,\underline{H}\,\underline{T}$.

Show that Bing, by picking second, can always win this game.  What is
the highest probability of winning that Bing can obtain if Anu makes
the best-possible first choice?

\problem A poker hand is made up of $5$ cards from an ordinary
52-card deck.  Give formulas for the probabilities of each of the
following hands:
\begin{itemize}

\item \defn{royal flush}

\item \defn{straight flush}

\item \defn{four of a kind}

\item \defn{full house}

\item \defn{flush}

\item \defn{straight}

\item \defn{three-of-a-kind}

\item \defn{two pair}

\item \defn{one pair}

\end{itemize}
Briefly explain each of your formulas, and calculate numerical values.

\problem In the game of contract bridge, an ordinary $52$-card deck is
dealt to four players, each receiving $13$ cards.  Players sitting
opposite each other are teammates.  After the bidding phase of the
game, the player winning the bid becomes the \defn{declarer} and his
partner becomes the \defn{dummy}; the players on the other team become
\defn{defenders}.  The defender sometimes has an easier time making
his contract if a given suit is split evenly between the defenders.
\begin{problemparts}
\ppart Compute the probability that if the defenders have five cards
of a suit between them, the cards are split $3:2$ or $2:3$ rather than
more unevenly.

\ppart Compute the probability that if the defenders have four cards
of a suit between them, the cards are split $2:2$ rather than more
unevenly.

\end{problemparts}

\end{problems}

\end{document}
