

\subsection{\bf Example of Bernoulli Trials}

\begin{slide}
\begin{itemize}
\item A course consists of 50 lectures given to 20 people, all of whom
get up before 9:00 a.m. and attend all classes.

\item A grand total of 1,000 smiles is observed.  

\item What is the chance that Frank Nagy (a physics graduate student
in times gone by) smiled exactly once during the whole course?

\item {\bf Solution:}\ Use previous formalism, setting each trial as a
smile: and success as Frank Nagy smiled.  

\item A priori, each smile has equal probability of belonging to each
person so $p=1/20$, $q=19/20$, and $n=1000$.

\item Thus: $Pr\{ \mbox{Frank Nagy smiles once and once only} \}$
\begin{eqnarray*}
& = & \left( \begin{array}{c} 1000 \\ 1 \end{array} \right) \cdot
\frac{1}{20} \cdot \left( \frac{19}{20} \right) ^{999} \\
& = & 50\, e^{999\log_e 19/20} \hspace{1em} \mbox{which isn't very big} \\
& \approx & 50\, e^{-51} \\
& \approx & 10^{-19}
\end{eqnarray*}
\end{itemize}
\end{slide}

