batch as the considered steel bar and that the test result is fˆy=350 MPa i.e. equal to the mean
value of the prior probabilistic model offy. Then the likelihoods are the probabilities of the
observation ˆfy given the event of failure and survival, respectively.
The likelihoods corresponding to the situation where the steel bar is not strengthened i.e.
decision a 0 , are calculated using e.g. FORM/SORM analysis, see Madsen et al. (1986).
(^122)
0
2
ˆ 0
ˆ 1.66 10
0
:
yy y
y
y
Pf f f A s
Pf
Pf As
(^122)
1
2
ˆ 0
ˆ 1.98 10
0
:
yyy
y
y
ffAs
Pf
Pf A s
Pf
where fy 1 and fy 2 are two different identical distributed random variables with distribution
function taken as the prior distribution for fy and with common parameters @ and @. A =
104 mm^2.
In the expressions for the calculation of the likelihoods the first event in the numerator is the
observation event. It is in this event where the modelling of the accuracy of the inspection or
test method must be included. In the above example no account of measurement uncertainty
was considered. Adding a random variable to the measured yield stress fˆycould have done
this. The more measurement uncertainty the weaker is the likelihood.
The posterior probabilities for the two states : 0 and : 1 may now be calculated using the prior
probabilities:
Pa@
: 00 11 .15 10^2 0.9885
and
Pa@
: 10 1.15 10 ^2
as
22
(^00222)
ˆ, 1.66 10 (1 10 ) 0.9905
(1.66 10 (1 1.15 10 1.98 10 1.15 )
Pfa: y
@@
^2 10
1.15
)
(^222)
10 222 10
a 1
2
ˆ, 1.98 10^10 0.95 10
(1.66 10 (1 1.15 10 1.98 10 1.15 )
Pfa: y
@@
1.15
)
By comparison with the prior probabilities it is readily seen that the test result has reduced the
probability of failure.
Now the posterior probabilities for the situation where the steel bar is strengthened by an
increase of the cross sectional area of 10% i.e. decision are considered. The calculations