Titel_SS06

Furthermore let the event A be an event in the sample space. Then the probability of the
eventA, i.e.PA() can be written as:

12

11 2 2

1

( ) ( ) ( ) ... ( )

( ) ( ) ( ) ( ) ... ( ) ( )

()()

n

n

n

ii

i

PA PA E PA E PA E

PAE PE PAE PE PAE PE

PAE PE









n (2.8)

this is also referred to as the total probability theorem.

From Equation (2.4) it isPAE PE( ii)( ) (PE APAi)() implying that:

()(

()

()

i

i

PAE PE

PE A

PA



i)

(2.9)

Now by inserting Equation (2.8) into Equation (2.9) the Bayes’ ruleresults:

1

()()

()

()(

ii

i n

jj)

j

PAE PE

PE A

PAE PE







(2.10)

In Equation (2.10) PE A( i ) denotes the posterior probability of Ei, the conditional term
P(A E )i is often referred to as the likelihood (i.e. the probability of observing a certain state
given the true state). The term is the prior probability of the event (i.e. prior to the
knowledge about the event

PE()i Ei

A).

As mentioned previously, the rule due to Bayes’ is extremely important, and in order to
facilitate the appreciation of this a few illustrative applications of Bayes’ rule will be given in
the following.

Example 2.1 – Using Bayes’ rule for concrete assessment

A reinforced concrete beam is considered. From experience it is known that the probability
that corrosion of the reinforcement has initiated (the event CI) is. However, in
order to know the condition more precisely an inspection method (non-destructive) has been
developed.

PCI( ) 0.01

The quality of the inspection method may be characterised by the probability that the
inspection method will indicate I initiated corrosion given that corrosion has initiated
PICI( ) (the probability of detection or equivalently the likelihood of an indication I given
corrosion initiation CI) and the probability that the inspection method will indicate initiated
corrosion given that no corrosion has initiated PICI( ) (the probability of erroneous findings
or the likelihood of an indication given no corrosion initiation).

For the inspection method at hand the following characteristics have been established:

PICI( )= 0.8

PICI( ) = 0.1