\problemdata      
        {?} % latex-friendly label for the prob.
        {Probability-Intro} % The topic(s) of the problem content.
        {?} % Source (if known)
        {F01, TU9; S00, Miniquiz 12} % Usage (list ones you are aware of).
        {Christos Kapoutsis, S02} % Last revision info (author, date).

\begin{problem}
Suppose a game involves tossing a fair coin until a head first arises.
\bparts
\ppart
What is the probability that a head first arises on an
even-numbered throw?

\solution{
The probability that a head first arises on an even-numbered throw can be
easily broken into two cases based on the first flip.  Half the time, a
head comes up on the first flip and thus the first head does not appear on
an even toss.  The other half of the time, the first head will be on an
even toss only if it is on a odd toss in the remaining sequence of tosses.
Thus:
\begin{eqnarray*}
p & = & \frac{1}{2}(1 - p) \\
p & = & \frac{1}{2} - \frac{1}{2}p \\
\frac{3}{2}p & = & \frac{1}{2} \\
p & = & \frac{1}{3} 
\end{eqnarray*}
}

\ppart What is a probability space for this experiment?

\solution{One reasonable probability space has as outcomes the strings
of the form $\mathtt{T}^k\mathtt{H}$, where $\mathtt{T}^k$ is a string of
$k$ \texttt{T}'s.  We define
\[
\pr{\mathtt{T}^k\mathtt{H}} \eqdef (\frac{1}{2})^{k+1}.
\]

To verify that this is a probability space, we must check that the sum of
the outcome probabilities is 1, namely,
\[
\sum_{k = 0}^{\infty} (\frac{1}{2})^{k+1} = 1.
\]
This follows straightforwardly from our formula for the sum of an infinite
geometric series.}

\ppart Does your probability space model the possibility of flipping an
infinite sequence of Tails?  If so, what is the probability of this event,
and what would happen if you removed this event?  If not, how do you
justify leaving it out?

\solution{ The probability space above does not have an event
corresponding to flipping Tails forever.  Nevertheless, it is a legitimate
probability space according to the formal definitions.  We could also add
an outcome or outcomes corresponding to flipping forever, but since the
probabilities of the events that end when the first Head is flipped
already total 1, the total probability of any flip-forever outcomes must
be zero.

How do we select the ``right'' probability space in this case?  In fact,
could it be that both of these spaces or wrong?  There is no mathematical
way to settle this question.  The spaces described here are mathematically
well-formed probability spaces, and ``rightness'' or ``wrongness'' is not
meant in a mathematical sense but in an empirical (experimental) sense.
Namely, can we use it to predict the results of actual coin flipping?

This is actually a subtle, and philosophically controversial, question,
since there is no experimental observation that could prove the model was
wrong.  For example, if in some experiment we flipped 1000 Tails before
flipping a Head, we still can't say that this \emph{proves} we have chosen
the wrong probability space, because the observation of 1000 tails is
\emph{possible} in our model, just very unlikely.  In practice, if we saw
1000 Tails in row, we would reject the model.  }

\iffalse

\ppart Does our probabilistic model make sense because of the
algebra, or the algebra makes sense because of our model?

\solution{ The algebra makes sense because of our model: Probability
is a tool to model real experiments. If two people were sitting on a
table flipping a coin forever we never know whether they would keep
bringing tails forever in reality, but we can model their game using a
probability space.  Within this probability space we assign
probabilities that make sense to us. Instead of claiming (falsely)
that TTH has probability 1/2, we are more rational and say ``Since my
coin is fair for each throw then I better assign $(1/2)^3$. Therefore
the algebra makes sense after we define a probability space.  }
\fi

\eparts

\end{problem}