

\begin{slide}
\subsection{\bf From Mathematical to Intuitive}
\subsection{\bf Definition of Probability}
{\Smallsize
\begin{itemize}
\item Anyhow, the variable point $X$ is now called a random variable
and $P(S)$ is the probability function of $X$.

\item To relate this definition to the outside world, we must resort to the
vanishing chalk dust approach as espoused by Mathews and Walker.  

\item If we flip an (unbiased) coin $N$ times, we expect
\[
\lim_{N\rightarrow\infty} \frac{N_h}{N} = \frac{1}{2}
\]
where $N_h$ is the number of times it turns up heads.  

\item We define
\[
P{\rm (heads)} = \lim_{N\rightarrow\infty} \frac{N_h}{N} = \frac{1}{2}
\]

\item This is a random variable with only two possible values --- heads
or tails.  Alternately, we could say it takes values $x=0$ and 1.

\item Then, in our previous language any Borel set containing 0 and not
1 has $P(S) = 1/2$, any containing 0 and 1, $P(S) = 1$, etc.

\item This illustrates the relation of intuitive frequency definition
of probability and that based on mathematical measure.
\end{itemize}
}
\end{slide}

