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Prof.~David Draper \\
Department of \\
\hspace*{0.1in} Applied Mathematics and Statistics \\
University of California, Santa Cruz
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\textbf{\large AMS 131: Quiz 2}
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Name: \underline{\hspace*{5.0in}}
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(You can use another page if necessary in any or all parts of the
problems.)
1.~Each year starts on one of the seven days (Sunday through Saturday). Each year is either a leap year (i.e., it includes February 29) or not. How many different calendars are possible for a year? Explain briefly.
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2.~A box contains $n$ balls, of which $r$ are red ($r$ and $n$ are both positive integers, and $r \le n$; suppose further that $n$ is even). Consider what happens when the balls are drawn from the box one at a time, at random \textbf{without replacement}. Determine
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\item[(a)]
the probability that the first ball drawn will be red;
\item[(b)]
the probability that the $\left( \frac{ n }{ 2 } \right)$th ball drawn will be red; and
\item[(c)]
the probability that the last ball drawn will be red.
\end{itemize}
Explain briefly in each case. \textit{Hint:} Imagine that the $n$ balls are randomly ordered in a list, and then drawn in that order, which is equivalent to sampling at random without replacement. \textit{Further Hint:} When choosing a sample $( Y_1, Y_2 )$ of size 2 at random without replacement from the population $\{ 1, 2, 9 \}$ in class recently, what was the relationship between $P ( Y_1 = 9 )$ and $P ( Y_2 = 9 )$?
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