Titel_SS06

different from structure to structure. In such cases the Bayesian interpretation of probability is
far more appropriate.

The basic idea behind the Bayesian statistics is that lack of knowledge should be treated by
probabilistic reasoning, similarly to other types of uncertainty. In reality, decisions have to be
made despite the lack of knowledge and probabilistic tools are a great help in that process.

2.3 Conditional Probability and Bayes’ Rule

Conditional probabilities are of special interest in risk and reliability analysis as they form the
basis of the updating of probability estimates based on new information, knowledge and
evidence.

The conditional probability of the event E 1 given that the event E 2 has occurred is written as:

12

12

2

(

()

()

PE E

PE E

PE



)

(2.4)

It is seen that the conditional probability is not defined if the conditioning event is the empty
set, i.e. whenPE()0 2 .

The event E 1 is said to be probabilistically independent of the event E 2 if :

PE E()( 12 PE 1 )

2

(2.5)

implying that the occurrence of the event E 2 does not affect the probability of E 1.

From Equation (2.4) the probability of the event EE 1
may be given as:

PE E()()( 12 

PE E PE12 2)

)

(2.6)

and it follows immediately that if the events E 1 and E 2 are independent, then:

PE E()()( 12 

PE PE1 2 (2.7)

Based on the above findings, the important Bayes’ rule can be derived.

Consider the sample space divided into mutually exclusive events n EE E 12 ,,..n (see also
Figure 2.1, where the case of n 8 is considered).

Ω

E 1 E 2

E 5 E 6 E 8

E 4

E 7

E 3

A Ω

E 1 E 2

E 5 E 6 E 8

E 4

E 7

E 3

A

Figure 2.1: Illustration of the rule of Bayes.