that is not a requirement. What is the probability that you will pull out a blue M&M? You
can all probably answer this question without knowing anything more about probability.
Because 24% of the M&M’s are blue, and because you are sampling randomly, the proba-
bility of drawing a blue M&M is .24. This example illustrates one definition of probability:
If an event can occur in Aways and can fail to occur in Bways, and if all possible ways
are equally likely (e.g., each M&M in the bag has an equal chance of being drawn),
then the probability of its occurrence is A/(A 1 B), and the probability of its failing to
occur is B/(A 1 B).
Because there are 24 ways of drawing a blue M&M (one for each of the 24 blue
M&M’s in a bag of 100 M&M’s) and 76 ways of drawing a different color, A 5 24, B 5 76,
and p(A) 5 24/(24 1 76) 5 .24.
An alternative view of probability is the frequentist view.Suppose that we keep draw-
ing M&M’s from the bag, noting the color on each draw. In conducting this sampling study
we sample with replacement,meaning that each M&M is replaced before the next one is
drawn. If we made a very large number of draws, we would find that (very nearly) 24% of
the draws would result in a blue M&M. Thus we might define probability as the limit^2 of
the relative frequency of occurrence of the desired event that we approach as the number
of draws increases.
Yet a third concept of probability is advocated by a number of theorists. That is the
concept of subjective probability.By this definition probability represents an individ-
ual’s subjective belief in the likelihood of the occurrence of an event. For example, the
statement, “I think that tomorrow will be a good day,” is a subjective statement of degree
of belief, which probably has very little to do with the long-range relative frequency of
the occurrence of good days, and in fact may have no mathematical basis whatsoever. This
is not to say that such a view of probability has no legitimate claim for our attention. Sub-
jective probabilities play an extremely important role in human decision-making and gov-
ern all aspects of our behavior. Just think of the number of decisions you make based on
subjective beliefs in the likelihood of certain outcomes. You order pasta for dinner be-
cause it is probably better than the mystery meat special; you plan to go skiing tomorrow
because the weather forecaster says that there is an 80% chance of snow overnight; you
bet your money on a horse because you think that the odds of its winning are better than
the 6:1 odds the bookies are offering. We will shortly discuss what is called Bayes’ theo-
rem, which is essential to the use of subjective probabilities. Statistical decisions as we
will make them here generally will be stated with respect to frequentist or analytical
approaches, although even so the interpretationof those probabilities has a strong subjec-
tive component.
Although the particular definition that you or I prefer may be important to each of us,
any of the definitions will lead to essentially the same result in terms of hypothesis testing,
the discussion of which runs through the rest of the book. (It should be said that those who
favor subjective probabilities often disagree with the general hypothesis-testing orienta-
tion.) In actual fact most people use the different approaches interchangeably. When we
say that the probability of losing at Russian roulette is 1/6, we are referring to the fact that
one of the gun’s six cylinders has a bullet in it. When we buy a particular car because
Consumer Reportssays it has a good repair record, we are responding to the fact that a high
proportion of these cars have been relatively trouble-free. When we say that the probabilitySection 5.1 Probability 113(^2) The word limitrefers to the fact that as we sample more and more M&M’s, the proportion of blue will get
closer and closer to some value. After 100 draws, the proportion might be .23; after 1000 draws it might be .242;
after 10,000 draws it might be .2398, and so on. Notice that the answer is coming closer and closer to
p 5 .2400000.... The value that is being approached is called the limit.
frequentist view
sample with
replacement
subjective
probability