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\begin{document}

\Large

\title{Introductory Example:  How Do Gambling Casinos Make Money?}

\author{}

\date{}

\maketitle

\pagestyle{fancy} 
\chead{}
\cfoot[]{I.\thepage:  Probability for Kids,
\copyright{2002}, N.  Matloff, University of California, Davis}


A simple game:

Roll one die ({\it dice} is the plural of {\it die}).

Here is what you win:

\vspace{0.3cm}
{\centering \begin{tabular}{|c|c|}
\hline 
if you roll...&
then you win\\
\hline 
\hline 
1, 2 or 3&
0\\
\hline 
4 or 5&
\$0.75\\
\hline 
6&
\$12.00\\
\hline 
\end{tabular}\par}\vspace{0.3cm}

{\bf Question:}  How much should the casino charge players to play this
game?

\newpage

\ 

Watch what happens as more and more people play the game:

Say 600 people have played the game so far.  Then:

\begin{itemize}

\item $\approx300$ people have won 0

\item $\approx200$ people have won \$0.75

\item $\approx100$ people have won \$12.00

\item so the casino has paid out $ \approx 200 \times \$0.75 + 100 \times
\$12.00$

\end{itemize}

So, the casino has paid out about \$1,350, for 600 players.

So, how much should the casino have charged as a fee to play this game,
in order for the casino to not lose money?

\newpage

\ 

How about charging \$1?  No, then the casino would have taken in $600
\times \$1 = \$600$, while paying out \$1,350 --- the casino would go
out of business!

How about charging \$3?  Then the casino takes in \$1,800 and doesn't
lose money, but a lot of people may consider \$3 too expensive, and not
play anymore.

To just ``break even,'' the casino would need to charge

$$
\mathrm{
\frac{\$1,350}{600} = \$2.25}
$$

(Then maybe charge a bit more than this, to cover other costs and add
profit, etc.) 

\newpage

\ 

What we did above was calculate something called {\it expected value}.
We'll return to this at a later date, but right now let's focus on the
relation of expected value to probability.

We use the notation P( ) to mean ``probability of.''  For example, if
you flip a coin, we say 

$$
P(heads) = \frac{1}{2}
$$

If you go back to our analysis of the casino, you will find that we made
use of the following probabilities:

$$
P(1, 2 ~ or ~ 3) = \frac{1}{2}
$$

$$
P(4 ~ or ~ 5) = \frac{1}{3}
$$

$$
P(6) = \frac{1}{6}
$$

So, probabilities are key to the casinos.  Thus we will
concentrate on probability for now, and return to expected value later.

\newpage

\ 

Homework:

(We will \underline{not} have a lot of homework, but we do need a little
bit each week, so that we don't forget the concepts.  Hand in your
solution in written form.  You may work with one partner if you wish, in
which case turn in just one paper with both of your names.)

The game is to toss a coin 3 times.  The player wins \$20 for 3 heads,
\$8 for 2 heads, and 0 otherwise.  Determine how much the casino must
charge to just break even.

Hint:  Use a 3-letter notation for the outcome.  TTT means all 3 flips
result in tails; TTH means the first two flips are tails and then the
third flip is heads; THT means the player gets a tail, then a head, then
a tail; etc., with 8 of these patterns in all.

Next week:

{\LARGE \bf The
Totally-Magical-and-All-Powerful-But-Seemingly-Nothing-to-It Principle of
Probability}

This is secret stuff!  No book and no other teacher will tell you this,
but it is extremely powerful.

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