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{\Large \textbf{Sample Questions: Counting Methods for Computing Probabilities}}%\footnote{}
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STA256 Fall 2018. Copyright information is at the end of the last page.
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\item Using the formula for $\binom{n}{r}$ from the formula sheet, and the Multiplication Principle, prove that the number of ways that $n$ objects can be divided into $r$ subsets with $n_i$ objects in set $i$ is $\binom{n}{n_1~\cdots~n_r}=\frac{n!}{n_1!~\cdots~n_r!}$.
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\item  Sample $r$ balls from a jar containing $n$ numbered balls. How many outcomes are there is the sampling is

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     \item With replacement? \vspace{100mm}
     \item Without replacement?
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\item  Using the formula for $_nP_r$ from the formula sheet, and the Multiplication Principle, prove $\binom{n}{r} = \frac{n!}{r! \, (n-r)!}$.
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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
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     \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, $k = 0, \ldots, 5$?
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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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\item In how many ways can 20 basketball players be divided into 4 teams of 5?  \vspace{100mm}

\item In how many ways can 6 red flags, 2 blue flags and 4 yellow flags be arranged? The flags are indistinguishable.
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\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.
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     \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?
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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/256f18} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f18}}
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