An inspection of the concrete beam is conducted indicating that corrosion has initiated. Based
on the findings from the inspection, what is the probability that corrosion of the reinforcement
has initiated?
The answer is readily found by applying the rule of Bayes’:
()() (
()
()()()() ()PICI PCI PI CI
PCI I
P I CI P CI P I CI P CI PI
)
(2.11)
With PI(), the probability of obtaining an indication of corrosion at the inspection:
PI( ) PICI PCI PICI PCI( ) ( ) ( ) ( ) 0.8 0.01 0.1 (1 0.01) 0.107and , the probability of achieving an indication of corrosion and at the same time to
have corrosion initiated:
PI CI()
PI CI(
) PICI PCI( ) ( ) 0.8 0.01 0.008
Thus, the probability that corrosion of the reinforcement has initiated given an indication of
corrosion by the inspection is:
( ) 0.008 0.075
0.107PCI I
The probability of initiated corrosion, given an indication of corrosion, is surprisingly low.
This is due to the high probability of an erroneous indication of corrosion at the inspection
relative to the small probability of initiated corrosion (i.e. the inspection method is not
sufficiently accurate for the considered application).
Example 2.2 – Using Bayes’ rule for bridge upgrading
An old reinforced concrete bridge is reassessed in connection with an upgrading of the
allowable traffic (see also Schneider, 1994). The concrete compressive strength class is
unknown but concrete cylinder samples may be taken from the bridge and tested in the
laboratory.
The following classification of the concrete is assumed:
B 1 : 03 c 0
B 2 : 30 c 40
B 3 : 40 cEven though the concrete class is unknown, experience with similar bridges suggests that the
probability of the concrete of the bridge belonging to class B1, B2 and B3 is 0.65, 0.24 and
0.11, respectively. This information comprises the prior information – prior to any experiment
result.
The test method is not perfect in the sense that even though the test indicates a value of the
concrete compressive strength belonging to a certain class, there is a certain probability that