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{\Large \textbf{Sample Questions: Counting Methods for Computing Probabilities}}%\footnote{}
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STA256 Fall 2019. Copyright information is at the end of the last page.
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\begin{enumerate} 

\item  Ten 
students are standing in line. If they lined up completely at random, what is the probability that Romeo is standing next to Juliet? This question was on the 2018 final exam. \vspace{100mm}

\item A deck of 52 cards is shuffled thoroughly. What is the probability that the four
aces are all next to each other?
\pagebreak

\item A standard deck of 52 cards has four ``suits:" spades, diamonds, hearts and clubs. Within each suit, the face values of the 13 cards are 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace. A ``hand" of poker is 5 cards, selected randomly without replacement.
\begin{enumerate}
     \item A ``flush" is a hand with 5 cards all of the same suit. What is the probability of a flush? \vspace{65mm}
     \item A ``straight" is a hand in which the 5 cards are in sequence. Suit is ignored. An Ace can be either high or low. What is the  probability of a straight?
\end{enumerate}
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\item Using the formula for $\binom{n}{k}$ from the formula sheet, and the Multiplication Principle, prove that the number of ways that $n$ objects can be divided into $k$ subsets with $n_i$ objects in set $i$ is $\binom{n}{n_1~\cdots~n_k}=\frac{n!}{n_1!~\cdots~n_k!}$.
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\item In how many ways can 20 basketball players be divided into 4 teams of 5?  \vspace{70mm}

\item In how many ways can 6 red flags, 2 blue flags and 4 yellow flags be arranged? The flags are indistinguishable.  \vspace{70mm}

\item In a game of poker, four players are each dealt 5 cards from a 52-card deck. How
many ways are there to deal the cards?
\pagebreak

\item  Sample $r$ balls from a jar containing $n$ numbered balls. How many outcomes are there is the sampling is

\begin{enumerate}
     \item With replacement? \vspace{100mm}
     \item Without replacement?
\end{enumerate}
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\item  A jar contains 10 red balls and 20 blue balls. If 5 balls are randomly sampled without replacement, what is the probability of
\begin{enumerate}
     \item All blue? \vspace{50mm}
     \item Two red and three blue? \vspace{50mm}
     \item At least one red? \pagebreak
     \item A jar contains 10 red balls and 20 blue balls. If 5 balls are randomly sampled without replacement, what is the probability of obtaining $k$ red balls, $k = 0, \ldots, 5$?
\end{enumerate}
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\item A shipment of $n$ electronic components has $k$ defectives. If we sample $m$ components without replacement, what is the probability of observing at least one defective?

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This assignment was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Mathematical and Computational Sciences, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website: 
     
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\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f19} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f19}}
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