Remarks on the Concept of "Probability"
Prerequisites
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Learning Objectives
 Define symmetrical outcomes
 Distinguish between frequentist and subjective approaches
 Determine whether the frequentist of subjective approach is better
suited for a given situation
Inferential
statistics is built on the foundation of probability theory,
and has been remarkably successful in guiding opinion about the
conclusions to be drawn from data. Yet (paradoxically) the very
idea of probability has been plagued by controversy from the beginning
of the subject to the present day. In this section we provide
a glimpse of the debate about the interpretation of the probability
concept.
One conception of probability is drawn from the
idea of symmetrical outcomes.
For example, the two possible outcomes of tossing a fair coin
seem not to be distinguishable in any way that affects which side
will land up or down. Therefore the probability of heads is taken
to be 1/2, as is the probability of tails. In general, if there
are N symmetrical outcomes, the probability of any given one of
them occurring is taken to be 1/N. Thus, if a sixsided die is
rolled, the probability of any one of the six sides coming up
is 1/6.
Probabilities can also be thought of in terms of
relative frequencies. If we tossed
a coin millions of times, we would expect the proportion of tosses
that came up heads to be pretty close to 1/2. As the number of
tosses increases, the proportion of heads approaches 1/2. Therefore,
we can say that the probability of a head is 1/2.
If it has rained in Seattle on 62% of the last
100,000 days, then the probability of it raining tomorrow might
be taken to be 0.62. This is a natural idea but nonetheless
unreasonable if we have further information relevant to whether
it will rain tomorrow. For example, if tomorrow is August 1,
a day of the year on which it seldom rains in Seattle, we should
only consider the percentage of the time it rained on August
1. But even this is not enough since the probability of rain
on the next August 1 depends on the humidity. (The chances are
higher in the presence of high humidity.) So, we should consult
only the prior occurrences of August 1 that had the same humidity
as the next occurrence of August 1. Of course, wind direction
also affects probability ... You can see that our sample of
prior cases will soon be reduced to the empty set. Anyway, past
meteorological history is misleading if the climate is changing?
For some purposes, probability is best thought of
as subjective. Questions such as "What is the probability
that Ms. Jones will defeat Mr. Smith in an upcoming congressional
election?" do not conveniently fit into either the symmetry
or frequency approaches to probability. Rather, assigning probability
0.7 (say) to this event seems to reflect the speaker's personal
opinion  perhaps his willingness to bet according to certain
odds. Such an approach to probability, however, seems to lose
the objective content of the idea of chance; probability becomes
mere opinion.
Two people might attach different probabilities
to the election outcome, yet there would be no criterion for calling
one "right" and the other "wrong." We cannot
call one of the two people right simply because she assigned higher
probability to the outcome that actually transpires. After all,
you would be right to attribute probability 1/6 to throwing a
six with a fair die, and your friend who attributes 2/3 to this
event would be wrong. And you are still right (and your friend
is still wrong) even if the die ends up showing a six! The lack
of objective criteria for adjudicating claims about probabilities
in the subjective perspective is an unattractive feature of it
for many scholars.
Like most work in the field, the present text adopts
the frequentist approach to probability. Moreover, almost all
the probabilities we shall encounter will be nondogmatic, that
is, neither zero nor one. An event with probability 0 has no chance
of occurring; an event of probability 1 is certain to occur. It
is hard to think of any examples of interest to statistics in
which the probability is either 0 or 1. (Even the probability
that the Sun will come up tomorrow is less than 1.)
The following example illustrates our attitude
about probabilities. Suppose you wish to know what the weather
will be like next Saturday because you are planning a picnic.
You turn on your radio, and the weather person says, “There
is a 10% chance of rain.” You decide to have the picnic
outdoors and, lo and behold, it rains. You are furious with
the weather person. But was she wrong? No, she did not say it
would not rain, only that rain was unlikely. She would have
been flatly wrong only if she said that the probability is 0
and it subsequently rained. However, if you kept track of her
weather predictions over a long period of time and found that
it rained on 50% of the days that the weather person said the
probability was 0.10, you could say her probability assessments
are wrong.
So when is it accurate to say that the probability
of rain is 0.10? According to our frequency interpretation, it
means that it will rain 10% of the days on which rain is forecast
with this probability.
