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\title{Counting Methods for Computing Probabilities\footnote{ This slide show is an open-source document. See last slide for copyright information.} \\ (Section 1.4)}
\subtitle{STA 256: Fall 2019}
\date{} % To suppress date
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% Text Section 1.4 title
\frametitle{Uniform Probability on Finite Spaces} \pause
% \framesubtitle{}
If the sample space is finite \pause and all outcomes of an experiment are equally likely, \pause
{\Large
\begin{eqnarray*}
P(A) &=& \frac{\mbox{Number of ways for $A$ to happen}}
{\mbox{Total number of outcomes}} \pause \\
&&\\
&=& \frac{|A|}{|S|}
\end{eqnarray*} \pause
} % End size
\vspace{8mm}
Need to count.
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\frametitle{A Boring Example} \pause
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Roll a fair die. What is the probability of an odd number? \pause
\begin{eqnarray*}
P(\mbox{Odd}) &=& \pause P(\{1,3,5\}) \pause \\
&&\\
&=& \frac{3}{6} \pause \\
&&\\
&=& \frac{1}{2}
\end{eqnarray*}
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\begin{frame}
\frametitle{Multiplication Principle}
\framesubtitle{Also called the Fundamental Principle of Counting} \pause
If there are $k$ experiments and the first has $n_1$ outcomes, the second has $n_2$ outcomes, etc., \pause then there are
\begin{displaymath}
n_1 \times n_2 \times \cdots \times n_k
\end{displaymath}
outcomes in all.
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\frametitle{Sample Question}
%\framesubtitle{}
If there are nine horses in a race, in how many ways can they finish first, second and third? \pause
\begin{displaymath}
9 \times 8 \times 7 = 504
\end{displaymath}
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\frametitle{Permutations: Ordered subsets}
\framesubtitle{The notation $_nP_k$ is not in the text} \pause
The number of \emph{permutations} (ordered subsets) of $n$ objects taken $k$ at a time is
\pause
\vspace{4mm}
{\Large
\begin{eqnarray*}
_nP_k & = & n \times (n-1) \times \cdots \times (n-k+1) \\ \pause
& = & \frac{n!}{(n-k)!}
\end{eqnarray*}
} % End size
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\frametitle{Combinations: Unordered subsets}
\framesubtitle{The vocabulary ``combinations" is not in the text} \pause
The number of \emph{combinations} (unordered subsets) of $n$ objects taken $k$ at a time is
\pause
\vspace{4mm}
{\Large
\begin{displaymath}
\binom{n}{k} = \frac{n!}{k! \, (n-k)!}
\end{displaymath}
} % End size
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\frametitle{Example}
% \framesubtitle{}
If a club has 30 members, how many ways are there to choose a committee of 5? \pause
%{\LARGE
\begin{eqnarray*}
\binom{30}{5} & = & \frac{30!}{5!(30-5)!} \\
& & \\
& = &142,506
\end{eqnarray*}
%} % End size
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\frametitle{Proof of $\binom{n}{k} = \frac{n!}{k! \, (n-k)!}$} \pause
% \framesubtitle{}
Choose an unordered subset of $k$ items from $n$. \pause Then place them in order. \pause By the Multiplication Principle, \pause
%{\LARGE
\begin{eqnarray*}
& & _nP_k = \pause \binom{n}{k} \times k! \\ \pause
& \Rightarrow & \frac{n!}{(n-k)!} = \binom{n}{k} \times k! \\ \pause
& \Rightarrow & \binom{n}{k} = \frac{n!}{k! \, (n-k)!}
\end{eqnarray*}
$\blacksquare$
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\frametitle{Example}
%\framesubtitle{}
How many ways are there to deal a hand of 5 cards from a standard deck of 52 cards? \pause
\begin{eqnarray*}
\binom{52}{5} \pause & = & \frac{52!}{5! \, 47!} \pause \\
& = & 2,598,960
\end{eqnarray*} \pause
\begin{itemize}
\item If you could inspect one hand per second, \pause
\item It would take a little over 30 days to examine them all. \pause
\item Working 24/7. \pause
\item The point is that these counting arguments allow you to ``count" objects that are so numerous you could not literally even look at them all.
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\frametitle{Binomial Theorem}
\framesubtitle{Binomial coefficients appear in the Binomial Theorem}
{\LARGE
\begin{displaymath}
(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^kb^{n-k}
\end{displaymath}
} % End size
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\frametitle{Multinomial Coefficients}
% \framesubtitle{Proposition C in the text}
The number of ways that $n$ objects can be divided into $\ell$ subsets with $k_i$ objects in set $i$, $i = 1, \ldots,\ell$ is
\vspace{2mm} \pause
{\LARGE
\begin{displaymath}
\binom{n}{n_1~\cdots~k_\ell}=\frac{n!}{k_1!~\cdots~k_\ell!}
\end{displaymath}
} % End size
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\frametitle{Multinomial Theorem}
\framesubtitle{Nice to know}
{\LARGE
\begin{displaymath}
(x_1 + \cdots + x_\ell)^n = \sum_{\mathbf{k}} \binom{n}{k_1~\cdots~k_\ell} x_1^{k_1} \cdots x_\ell^{k_\ell}
\end{displaymath}
} % End size
where the sum is over all non-negative integers $k_1, \ldots, k_\ell$ such that $\sum_{j=1}^\ell k_j = n$.
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\frametitle{Copyright Information}
This slide show was prepared by \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistical 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:
\vspace{5mm}
\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f19} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f19}}
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