Environmental Engineering FOURTH EDITION

RiskAnalysis 37

The sum of the probabilities of each of these ten outcomes is unity. Therefore, only

nine of the ten probabilities are independent.

A more precise representation of these probabilities is that some are independent,

or unconditional, and some are conditional. Probabilities 2 through 10 are conditional

on the car being in an accident. A more precise representation of the sum is then

P(no accident) + P(accident) = 1

P(accident) = P(2) + P(3) + P(4) + P(5) + P(6) + P(7)

+ P(8) + P(9) + P(10) = 1 - P(no accident).

A conditional probability is the product of the probabilities. If the probability of an
accident is 10% (or O.l), and the probability of an injury to either of two occupants of
the vehicle conditional on that accident is 10% (or O.l), then the overall probability of
an injury to either occupant is

P(nonfatal injury) = P(accident) x P( injurylaccident) = 0.1 x 0.1 = 0.01

or 1%. The sum of the conditional probabilities must also be unity. Thus if the con-

ditional probability of an injury is 1% (0.01), then the conditional probability of an

accident involving no injury is 1 - 0.01 = 0.99. The overall probability of an accident

with neither an injury nor a death is then 0.1 x 0.99 = 0.099, or 9.9%.

This problem may be illustrated using an event tree, as shown in Fig. 3-1, where

the net probabilities are listed on the right-hand side of the figure. Note that these are

Driver killed,occupant unhurt (6)

Driver injured, occupant killed (7)

\\u

P=O.OOl

P=0.002

P=0.0036

P=0.0024

P=0.005

P=0.015

P=O.Ol

P=O.OOl

\ Accident, no injuries (2) -0.06

Car arrives at destination without incident (1) P=O.9

Figure 3-1. Event tree for automobile accident example, Numbers in parentheses refer

to the scenarios on page 36.