# 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.

1. INTRODUCTION

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

2. QUALITATIVE ASPECTS OF THE NOTION

OF RISK

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

92714.

2Pickard,Lowe and Ganick, Inc.

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0272-4332/Sl/0300-00I

1$03.00/1 01981 Society for Risk Analysis

Kaplan and Garrick

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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

+damage.

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 =

hazard

safeguards

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

small.

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

basis.

3. QUANTITATIVE DEFINITION OF RISK

(FIRST LEVEL)

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

13

Table 11. Scenario List with Cumulative Probability

Table I. Scenario List

~~

Consequence

Scenario Likelihood Consequences Cumulative probability

Scenario

Likelihood

Sl

PI

XI

SI

PI

XI

s2

P2

x 2

s2

P2

x 2

s,

P N

XN

s,

Pi

sN-l

pN-I

SN

P N

fore, a risk analysis consists of an answer to the

following three questions:

(i) What can happen? (i.e., What can go

wrong?)

(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.

5

9

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:

R={(si,pi,xi)},

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

of.

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.

4. 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

confidence.

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

18

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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