Uncertainty, Probability, and Expected Value 501assess the likelihood of their occurrence. Uncertainty is acknowledged in
expressions such as “it is likely,” “the odds are,” and “there is an outside
chance.” The difficulty with such qualitative expressions is that they are ambigu-
ous and open to different interpretations. One is prompted to ask, “How likely
is likely?” The essential means for quantifying statements of likelihood is to use
probabilities.It is far more useful for a meteorologist to state that there is a 60
percent chance of rain tomorrow than to claim that rain is likely. Probability has
been described as the mathematical language of uncertainty. The key is to have
a sound understanding of what probabilities mean.
The probabilityof an outcome is the odds or chance that the outcome will
occur. In the usual parlance, we speak of probabilities as ranging between 0
and 1. (An event having a probability of 1 is a certainty; an event having a prob-
ability of 0 is deemed impossible.) Whatever the probability, the relevant ques-
tion is: What is the basis for this assessment? Frequently there is an objective
foundation for the probability assessment. The chance of heads on a single toss
of a fair coin is 50 percent, or one-half. In a random draw, the chance of pick-
ing the lone black ball from a hat containing five balls is one in five, and so on.
When viewed closely, the main basis for assessments such as these is the
notion of a probability as a long-run frequency.If an uncertain event (like a coin
toss or a random draw) is repeated a very large number of times, the frequency
of the event is a measure of its true probability. For instance, if a fair coin is
tossed 1,000 times, the frequency of heads (i.e., the number of heads divided
by the total number of tosses) will be very close to .5. If the actual long-run fre-
quency turned out to be .6, we would be justified in asserting that the coin was
unfair. The frequency interpretation applies to most statistical data. For exam-
ple, if annual employment in the mining industry totals 40,000 workers and
80 workers die in mining accidents each year, the annual probability of a rep-
resentative mine worker dying on the job is 80/40,000 or .2 percent.
It should be evident that in many (and perhaps most) situations, there is no
chance that a situation will be repeated and therefore no way to assess probabil-
ities on frequency grounds. In its development of a new product (one that is
unique to the marketplace), a firm knows that the product launch is a one-shot
situation. The firm may believe there is a 40 percent chance of success, but there
is no way to validate this by launching the product 100 times and watching for 40
successes. Similarly, a company about to enter into patent litigation faces the
problem of predicting the likely outcome of a one-time legal suit. Still another
example is a business economist attempting to put odds on the likelihood of a
new oil price “shock” (say, a 50 percent rise in oil prices) over the next 18 months.
In dealing with such situations, decision makers rely on a subjectivenotion
of probability. According to the subjectiveview, the probability of an outcome
represents the decision maker’s degree of belief that the outcome will occur.
This is exactly the meaning of a statement such as “The chance of a successful
product launch is 60 percent.” Of course, in making a probability assessment,
the manager should attempt to analyze and interpret all pertinent evidence andc12DecisionMakingunderUncertainty.qxd 9/29/11 1:34 PM Page 501