Application Example 2
(Total Probability Theorem)
EVALUATION OF NATURAL AND MAN-MADE RISKS
An important application area of probability and statistics is the assessment of natural
and man-made risks. For example, one may need to evaluate the safety of an engineering
facility against extreme environmental actions, such as earthquakes, strong winds,
extreme floods, ocean waves, etc. Since environmental loads vary in time, one usually
expresses reliability through the probability that some undesirable “failure” event (severe
structural damage or collapse, levee breach, dam overtopping, ship hull buckling, etc.)
occurs at least once during a reference period of time T, for example 50 or 100 years.
In order to quantify
risk
, one needs to combine two elements:
1. A description of the severity of the environment, in terms of the probability P[L
T
> l]
with which the maximum environmental load in T years, L
T
, exceeds various levels
l
.
Evaluating P[L
T
>
l
] is often referred to as
hazard assessment
;
2. A description of the resistance of the facility in terms of the dependence of the
probability of system failure P
f
on the magnitude
l
of the environmental load. This
function, P
f
(
l
), is often referred to as the
fragility function
.
Once quantified, the hazard and fragility functions are combined to produce the
probability of (at least one) failure in T. This is done by using the
Total Probability
Theorem
, which says that, if {B
1
,..., B
n
} is a set of mutually exclusive and collectively
exhaustive events and A is any other event, then the probability of A can be calculated as