%% ----------------------------------------------------------------

In probability, one starts with a probability model describing what events we
think are going to occur, with what likelihoods.  The events may be random, in
the sense that we don't know for sure what will happen next, but we do quantify
our degree of surprise when various things happen.

The standard (maybe overused) example is flipping a fair coin.
``Fair'' means, technically, that the probability of heads on a given
flip is 50\%, and the probability of tails on a given flip is 50\%.  This
doesn't mean that every other flip will give a head --- after all, three
heads in a row is no surprise.  Five heads in a row would be more surprising,
and when you've seen twenty heads in a row you're sure that something fishy is
going on.  What the 50\% probability of heads does mean is that, as the number
of flips increases, we expect the number of heads to approach half the number
of flips.  Seven heads on ten flips is no surprise; 700,000 heads on 1,000,000
tosses is highly unlikely.

As this example illustrates, the probabilist starts with a probability model
(something which assigns various percentage likelihoods of different things
happening), then tells us which things are more and less likely to occur.

Key points about probability:

\begin{itemize}

\item Rules $\longrightarrow$ data:  Given the rules, describe the likelihoods
of various events occurring.

\item Probability is about prediction --- looking forward.

\item Probability is mathematics.

\end{itemize}

%% ----------------------------------------------------------------

The statistician turns this around:

\begin{itemize}

\item Rules $\longleftarrow$ data:  Given only the data, try to guess what the
rules were.  That is, some probability model controlled what data came out, and
the best we can do is guess what that model was.  We might guess wrong; we
might refine our guess as we get more data.


\item Statistics is about looking backward.


\item Statistics is an art.  It uses mathematical methods, but it is more than
math.


\item Once we make our best <i>statistical</i> guess about what the probability
model is (what the rules are), based on looking <i>backward</i>, we can then
use that <i>probability</i> model to predict the <i>future</i>.  (This is,
in part, why I say that probability doesn't need statistics, but statistics
uses probability.)

\end{itemize}

Here's my favorite example to illustrate.  Suppose I give you a list of heads
and tails.  You, as the statistician, are in the following situation:

\begin{itemize}

\item You do not know ahead of time that the coin is fair.  Maybe you've
  been hired to decide whether the coin is fair (or, more generally, whether a
  gambling house is committing fraud).


\item You may not even know ahead of time whether the data come from a
  coin-flipping experiment at all.

\end{itemize}

Suppose the data are three heads.  Your first guess might be that a fair coin
is being flipped, and these data don't contradict that hypothesis.  Based on
these data, you might hypothesize that the rules governing the experiment are
that of a fair coin:  your probability model for predicting the future is that
heads and tails each occur with 50\% likelihood.

If there are ten heads in a row, though, or twenty, then you might start to
reject that hypothesis and replace it with the hypothesis that the coin has
heads on both sides.  Then you'd predict that the next toss will certainly be
heads: your new probability model for predicting the future is that heads occur
with 100\% likelihood, and tails occur with 0\% likelihood.

If the data are ``heads, tails, heads, tails, heads, tails'', then
again, your first fair-coin hypothesis seems plausible.  If on the other hand
you have heads alternating with tails not three pairs but 50 pairs in a row,
then you reject that model.  It begins to sound like the coin is not being
flipped in the air, but rather is being flipped with a spatula.  Your new
probability model is that if the previous result was tails or heads, then the
next result is heads or tails, respectively, with 100\% likelihood.
