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Let us come full circle and reconsider the original problem posed at the beginning of this paper.

Consider the case of rolling a single die. A theoretician picks up the die, examines it, and makes the following statement: "The die has six sides, each side is equally likely to turn up, therefore the probability of any one particular side turning up is 1 out of 6 or 1/6. Furthermore, each throw of the die is completely independent of all previous throws." With that the theoretician puts the die down and leaves the room without ever having thrown it once.

An experimentalist, never content to accept what a cocky theoretician states as fact, picks up the die and starts to roll it. He rolls the die twelve times and records the outcome of each roll. His results are:

{ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 }

We now ask the question, "What is the probability of throwing a 1 on the next roll?"

There are three answers to this question. Each answer depends on what we believe to be
the true properties of the chance experiment.

**1. THE THEORETICIAN'S ANSWER**

The theoretician claims the above result is a legitimate possible outcome according to his model of
the chance experiment, as equally likely to occur as any of the other 6^{12} =
2,176,782,336 possible outcomes of rolling a six sided die twelve times. We may choose to
believe the theoretician's proclamation that each side of the die is still equally likely
to show up, and each throw of the die is still completely independent of all previous
throws. In this case the results of the previous twelve throws tell us nothing about what
the next throw might be, and the probability of the next throw being a 1 is still 1/6 by
definition.

The theoretician may support his argument by citing the physics of the experiment and the
great uncertainty of the initial condition of the die while being shaken in the hand.

**2. THE EXPERIMENTALIST'S ANSWER**

The experimentalist, never content to accept what a cocky theoretician states as fact, feels that we should take into account the previous results of rolling the die and reconsider the theoretician's proposed model of the chance experiment. Perhaps the die is actually unfair, heavily weighted in a lopsided way as to render the throwing of a 1 quite likely, and the throwing of any other number quite unlikely. The experimentalist comes up with a new model of the chance experiment, claiming the probability of throwing a 1 is near 100%, and the probability of throwing any other number is extremely low. We may then choose to abandon the theoretician's initial model of the chance experiment and refer to the new updated model of the chance experiment based on recent past experience. In this case the new model declares the probability of throwing a 1 on the next roll to be near 100% certain.

The experimentalist may support his argument that a new model of the chance experiment
is justified by performing the analysis presented in Chapter 2.

**3. THE COMPUTER PROGRAMMER'S ANSWER**

At this point a computer programmer enters the room and declares the die isn't a real die
after all but merely a computer simulation of a die. The computer uses a psuedo-random
number generator algorithm which in the long run simulates well a sequence of random
numbers—and that is the catch right there which throws off the probability of
throwing a 1 on the next roll. If we believe beforehand that our chance experiment is one
which in the long run will *appear* to be random, then we can claim the probability
of the next simulated throw of the die being a 1 is less than 1/6, because it's high time
some other number showed up. This is equivalent to the die having a memory of previous
results and adjusting it's next outcome accordingly to maintain the *appearance* of
randomness.

So we have three answers to the question, "What is the probability of throwing a 1 on
the next throw."

1. The probability is 1/6 because we believe the theoretician.

2. The probability is greater than 1/6 because we believe the experimentalist.

3. The probability is less than 1/6 because we believe the computer programmer.

Much confusion about probability theory results from failing to comprehend and understand
the above three possibilities. It all has to do with what we choose to believe are the
properties of the chance experiment.

Chapter 11 added September 2003

Chapter 10 | Conclusions | |||||||

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