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\title{General Mathematics and Computational Science I}
\author{Exercise 12}
\date{October 20, 2005}
\maketitle

\begin{enumerate}

\item A card is drawn at random from a standard deck of $52$ cards.
What is the probability of drawing
\begin{enumerate}
\item a king of spades?
\item a king?
\item not a king?
\item a diamond?
\item a face card (jack, queen, or king)?
\end{enumerate}

\item In the game of Yahtzee (or Yacht), five dice are thrown.  Show
that the probability of throwing a \emph{large straight} ($5$ numbers
in a row, the order does not matter) is $\tfrac{5}{162}$.

Alternatively, you may solve the following, harder, problem.  Show
that the probability of throwing a \emph{small straight} ($4$ numbers
in a row) is $\tfrac{10}{81}$.  Do not count small straights which are
also large straights.

(The actual rules of the game allow to repeat throwing a selected
subset of dice twice.  This is considerably more complicated to
analyze.) 

\item Show that 
\begin{equation*}
  \sqrt{a^2 + b^2} \leq a + b \leq \sqrt{2(a^2+b^2)}
\end{equation*}
for arbitrary non-negative real numbers $a$ and $b$.


\end{enumerate}
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