Remarks on the Concept of "Probability"
- Define symmetrical outcomes
- Distinguish between frequentist and subjective approaches
- Determine whether the frequentist or subjective approach is better
suited for a given situation
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
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 six-sided die is
rolled, the probability of any one of the six sides coming up
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. Garcia 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 in most cases.
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.
Please answer the questions: