2
Sample Space and Probability
Chap. 1
“Probability” is a very useful concept, but can be interpreted in a number of
ways. As an illustration, consider the following.
A patient is admitted to the hospital and a potentially life-saving drug is
administered. The following dialog takes place between the nurse and a
concerned relative.
RELATIVE
: Nurse, what is the probability that the drug will work?
NURSE
: I hope it works, we’ll know tomorrow.
RELATIVE
: Yes, but what is the probability that it will?
NURSE
: Each case is different, we have to wait.
RELATIVE
: But let’s see, out of a hundred patients that are treated under
similar conditions, how many times would you expect it to work?
NURSE
(somewhat annoyed): I told you, every person is different, for some
it works, for some it doesn’t.
RELATIVE
(insisting): Then tell me, if you had to bet whether it will work
or not, which side of the bet would you take?
NURSE
(cheering up for a moment): I’d bet it will work.
RELATIVE
(somewhat relieved): OK, now, would you be willing to lose two
dollars if it doesn’t work, and gain one dollar if it does?
NURSE
(exasperated): What a sick thought! You are wasting my time!
In this conversation, the relative attempts to use the concept of probability
to discuss an
uncertain
situation. The nurse’s initial response indicates that the
meaning of “probability” is not uniformly shared or understood, and the relative
tries to make it more concrete. The first approach is to define probability in
terms of
frequency of occurrence
, as a percentage of successes in a moderately
large number of similar situations. Such an interpretation is often natural. For
example, when we say that a perfectly manufactured coin lands on heads “with
probability 50%,” we typically mean “roughly half of the time.” But the nurse
may not be entirely wrong in refusing to discuss in such terms. What if this
was an experimental drug that was administered for the very first time in this
hospital or in the nurse’s experience?
While there are many situations involving uncertainty in which the fre-
quency interpretation is appropriate, there are other situations in which it is
not. Consider, for example, a scholar who asserts that the Iliad and the Odyssey
were composed by the same person, with probability 90%. Such an assertion
conveys some information, but not in terms of frequencies, since the subject is
a one-time event. Rather, it is an expression of the scholar’s
subjective be-
lief
. One might think that subjective beliefs are not interesting, at least from a
mathematical or scientific point of view. On the other hand, people often have
to make choices in the presence of uncertainty, and a systematic way of making
use of their beliefs is a prerequisite for successful, or at least consistent, decision
making.