HLSS 505 AMU Benefits of Risk-Based Approaches What are the benefits of risk-based approaches?When might an examination of only one risk factor be appropri

HLSS 505 AMU Benefits of Risk-Based Approaches What are the benefits of risk-based approaches?When might an examination of only one risk factor be appropriate for decision-making?When might only reviewing one risk factor lead to poor results? Risk Analysis, Vol. I , No. I , 1981
On The Quantitative Definition of Risk
Stanley Kaplan’ and B. John Garrick2
Received July 14, 1980
A quantitative definition of risk is suggested in terms of the idea of a “set of triplets.” The
definition is extended to include uncertainty and completeness, and the use of Bayes’ theorem
is described in this connection. The definition is used to discuss the notions of “relative risk,”
“relativity of risk,” and “acceptabilityof risk.”
KEY WORDS risk; uncertainty; probability; Baye’s theorem; decision.
quantitative definition. Since the notion of “probability” is fundamentally intertwined with the definition
of risk, the next section addresses the precise meaning adopted in this paper for the term “probability.”
In particular, at this point, we carefully draw a
distinction between “probability” and “frequency.”
Then, using this distinction, we return to the idea of
risk, and give a “second-level” definition (of risk
which generalizes the first-level definition) and is
large enough and flexible enough to include at least
all the aspects and subtleties of risk that have been
encountered in the authors’ experience.
As readers of this journal are well aware, we are
not able in life to avoid risk but only to choose
between risks. Rational decision-making requires,
therefore, a clear and quantitative way of expressing
risk so that it can be properly weighed, along with all
other costs and benefits, in the decision process.
The purpose of this paper is to provide some
suggestions and contributions toward a uniform conceptual/linguistic framework for quantifying and
making precise the notion of risk. The concepts and
definitions we shall present in this connection have
shown themselves to be sturdy and serviceable in
practical application to a wide variety of risk situations. They have demonstrated in the courtroom and
elsewhere the ability to improve communication and
greatly diminish the confusion and controversy that
often swirls around public decision making involving
risk. We hope therefore with this paper to widen the
understanding and adoption of this framework, and
to that end adopt a leisurely and tutorial place.
We begin in the next section with a short discussion of several qualitative aspects of the notion of
risk. We then proceed to a first-pass or first-level
The subject of risk has become very popular in
the last few years and is much talked about at all
levels of industry and government. Correspondingly,
the literature on the subject has grown very large [see
for example refs. (1-3)]. In this literature the word
“risk” is used in many different senses. Many different kinds of risk are discussed: business risk, social
risk, economic risk, safety risk, investment risk, military risk, political risk, etc. Now one of the requirements for an intelligible subject is a uniform and
consistent usage of words. So we should like to begin
sorting things out by drawing some distinctions in
‘Kaplan & Associates, Inc., 17840 Skypark Blvd. Irvine, CA
2Pickard,Lowe and Ganick, Inc.
1$03.00/1 01981 Society for Risk Analysis
Kaplan and Garrick
meaning between various of these words as we shall
use them. We begin with “risk” and “uncertainty.”
reduces risk. Thus, if we know there is a hole in the
road around the corner, it poses less risk to us than if
we zip around not knowing about it.
2.1. The Distinction Between Risk and Uncertainty
2.3. Relativity of Risk
Suppose a rich relative had just died and named
you as sole heir. The auditors are totaling up his
assets. Until that is done you are not sure how much
you will get after estate taxes. It may be $1 million or
$2 million. You would then certainly say you were in
a state of uncertainty, but you would hardly say that
you were facing risk. The notion of risk, therefore,
involves both uncertainty and some kind of loss or
damage that might be received. Symbolically, we
could write this as:
risk= uncertainty
This equation expresses our first distinction. As a
second, it is of great value to differentiate between
the notions of “risk” and “hazard.” This is the subject of the next section.
2.2. The Distinction Between Risk and Hazard
It is very useful, especially in understanding the
public controversies surrounding energy production
and transport facilities, to draw a distinction between
the ideas of risk and hazard.
In the di~tionary‘~)
we find hazard defined as “a
source of danger.” Risk is the “possibility of loss or
injury” and the “degree of probability of such loss.”
Hazard, therefore, simply exists as a source. Risk
includes the likelihood of conversion of that source
into actual delivery of loss, injury, or some form of
damage. This is the sense in which we use the words.
As an example, the ocean can be said to be a hazard.
If we attempt to cross it in a rowboat we undergo
great risk. If we use the Queen Elizabeth, the risk is
small. The Queen Elizabeth thus is a device that we
use to safeguard us against the hazard, resulting in
small risk. As in Sec. 2.1., we express this idea
symbolically in the form of an equation:
risk =
This equation also brings out the thought that we
may make risk as small as we like by increasing the
safeguards but may never, as a matter of principle,
bring it to zero. Risk is never zero, but it can be
Included under the heading “safeguards” is the
idea of simple awareness. That is, awareness of risk
Connected to this thought is the idea that risk is
relative to the observer. We had a case in Los Angeles
recently that illustrates this idea. Some people put a
rattlesnake in a man’s mailbox. Now if you had asked
that man: “Is it a risk to put your hand in your
mailbox?” He would have said, “Of course not.” We
however, knowing about the snake, would say it is
very risky indeed.
Thus risk is relative to the observer. It is a
subjective thing- it depends upon who is looking.
Some writers refer to this fact by using the phrase
“perceived risk.” The problem with the phrase is that
it suggests the existence of some other kind of riskother than perceived. It suggests the existence of an
“absolute risk.” However, under attempts to pin it
down, the notion of absolute risk always ends up
being somebody else’s perceived risk. This brings us
in touch with some fairly deep philosophical matters,
which incidentally are reminiscent of those raised in
Einstein’s theory of the relativity of space and time.
This subject will become clear after we have
given precise, quantitative definitions of “risk” and
“probability.” We begin this process in the next
section by giving the definition of risk. We postpone
the definitions of probability until Sec. 4. This order
of presentation departs a little from the logical order
because the definition of risk uses the term probability. This works out all right, however, since the reader
already has a good intuitive grasp of the meaning of
probability. The earlier discussions of risk will then
serve to motivate the detailed attention given to the
subtleties of the definition of probability.
So, qualitatively, risk depends on what you do
and what you know and what you do not know. Let
us proceed now to put the idea on a quantitative
3.1. “Set of Triplets Idea”
In analyzing risk we are attempting to envision
how the future will turn out if we undertake a certain
course of action (or inaction). Fundamentally, there-
On the Quantitative Definition of Risk
Table 11. Scenario List with Cumulative Probability
Table I. Scenario List
Scenario Likelihood Consequences Cumulative probability
x 2
x 2
fore, a risk analysis consists of an answer to the
following three questions:
(i) What can happen? (i.e., What can go
(ii) How likely is it that that will happen?
(iii) If it does happen, what are the consequences?
To answer these questions we would make a list of
outcomes or “scenarios” as suggested in Table I. The
ith line in Table I can be thought of as a triplet:
(SI PI xi )
where si is a scenario identification or description;
p, is the probability of that scenario; and
x, is the consequence or evaluation measure of
that scenario, i.e., the measure of damage.
If this table contains all the scenarios we can
think of, we can then say that it (the table) is the
answer to the question and therefore is the risk. More
formally, using braces, { }, to denote “set of” we can
say that the risk, R , “is” the set of triplets:
i=1,2 ,…, N .
This definition of risk as a set of triplets is our
first-level definition. We shall refine and enlarge it
later.3 For now let us show how to give a pictorial
representation of risk.
3.2. Risk Curves
Imagine now, in Table I, that the scenarios have
been arranged in order of increasing severity of
damage. That is to say, the damages x i obey the
ordering relationship:
x i D).
which includes all the scenarios we have thought of,
and also an allowance for those we have not thought
Thus extended, the set of scenarios may be said
to be logically complete.
It seems at first glance that what we have done
here is simply a logical trick which does not address
the fundamental objection. It is a little bit more than
a trick, however. For one thing, it takes the argument
out of the verbal realm and into the quantitative
realm. Instead of the emotional question, “What
about the things that you have not thought of?”
“What probability should we assign to the residual
category sN+
Once the question has been phrased in this way,
we can proceed like rational people, in the same way
we do to assign any probability. We ask what evidence do we have on this point? What knowledge,
what relevant experience? In particular, we note that
one piece of evidence is always present-namely that
scenarios of the type s N + l have not occurred yet,
otherwise we would have included them elsewhere on
the list.
How much is this piece of evidence worth? This
is a question that can be answered rationally within
the framework of the theory of probability using
Bayes’ theorem. We shall return to this point in Sec.
6. It is timely now to explain the sense in which we
are using the word probability.
People have been arguing about the meaning of
probability for at least 200 years, since the time of
Laplace and Bayes. The major polarization of the
argument is between the “objectivist” or “frequentist”
school who view probability as somethng external,
the result of repetitive experiments, and the “subjectivists” who view probability as an expression of
an internal state-a state of knowledge or state of
In this paper we adopt the point of view that
both schools are right; they are just talking about two
different ideas. Unfortunately, they both use the same
word-which seems to be the source of most of the
confusion. We shall, therefore, assign each idea the
dignity of its own name.
4.1. The Definition of Probability and Distinction
Between Probability and Frequency
What the objectivists are talking about we shall
call “frequency.” What the subjectivists are talking
about we shall call “probability.” Thus, “probability”
as we shall use it is a numerical measure of a state of
knowledge, a degree of belief, a state of confidence.
“Frequency” on the other hand refers to the outcome
of an experiment of some kind involving repeated
trials. Thus frequency is a “hard” measurable number. This is so even if the experiment is only a
thought experiment or an experiment to be done in
the future. At least in concept then, a frequency is a
well-defined, objective, measurable number.
Probability, on the other hand, at first glance is
a notion of a different kind. Defined, essentially, as a
number used to communicate a state of mind, it thus
seems “soft” and changeable, subjective- not measureable, at least not in the usual way.
The cornerstone of our approach is the idea that
given two meaningful statements (or propositions or
events), it makes sense to say that one is more (less,
Kaplan and Garrick
equally) likely than the other. That is, we accept as an
axiom the comparability of uncertainty. Since two
uncertain statements can be compared, the next logical step is to devise a scale to calibrate uncertainty.
This can be done in several ways. The most
direct, however, is to use frequency in the following
way.5 Suppose we have a lottery basket containing
coupons numbered from 1 to 1000. Suppose the
basket is to be thoroughly mixed, and that you are
about to draw a coupon blindfolded. We ask: Will
you draw a coupon numbered 632 or less? With
respect to t h s question you experience a certain state
of confidence. Similarly, I experience a state of confidence with respect to this same question. Let us agree
to call thts state of confidence, “probability 0.632.”
Now we …
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