March 15, 1999
CIVL 281 Mid-Semester Examination I
Problem 1 (35%)
A slope at a given site was studied by engineers who conclude that it could fail in
three different ways namely: a deep-seated slide (Event D) due to weak soil, or a
shallow slide (Event S) due to infiltration of rain, or washout failure (Event W) due to
scouring by running surface water. The probability of event D is only 0.001 whereas
the probabilities of events S and W are 0.02 and 0.01. The events S and W are related
such that if the slope did not have a shallow slide, the probability that it will not have
a washout failure is increased to 0.998. Suppose deep-seated slide can be statistically
independent of the other two events. Determine the following:
a. Probability of failure of the slope.
b. If the damages corresponding to the events D, S and W are 50, 20 and 10 million
dollars, what is the expected damage from failure of the slope?
c. To improve the safety of the slope, two options are considered:
I.
Provide a better drainage system
II.
Construct a hard cover on the slope
Option I is expected to reduce the probabilities of S and W by half. However
there is a 20% probability that the drainage system may not perform as expected;
in this case, no improvement in the probabilities will result. The performance of
Option II however may be assumed certain to eliminate the rainwater-induced
failures of events E and W.
i. What would be the reduction in the expected damages corresponding to
Options I and II respectively?
ii. What would you suggest as the best alternative if the costs of Options I and
II are 0.3 and 0.4 million dollars respectively?
d. Suppose an ultrasonic device can be used to detect the presence of weak soil that
may cause a deep-seated slide. The reliability of the device is such that it fails to
detect a weak soil layer 30% of the time and yet it gives false signal of the
presence of weak soil layer 20% of the time. Suppose the engineer initially
believes that the probability of presence of weak soil layer at the site is 1%. What
would be his updated probability if indeed the ultrasonic inspection reveals no
weak soil layer?
Problem 2 (35%)
To manage the speeding problem on Route 3 (R-3), the police sets up a radar station
to detect violations occasionally. However, due to budget constraints, they only do
that 1% of the time. Moreover, the radar gun used by the police is not accurate. The
probability of detecting a speeding vehicle is 95%; whereas it has a 2% probability of
charging an innocent vehicle (which is not speeding). It is known that 100,000
vehicles pass through R-3 per month and that 60% of them are speeding.
Determine:
a. the probability of getting a speeding ticket on R-3
b. the probability that a driver who receives a speeding ticket is actually innocent
c. the average number of tickets per month issued by the police on R-3
Subsequently, a new radio station broadcasts the locations of the police radar stations
regularly. However, this information is not perfect. When the police sets up a radar
station on R-3, it has a probability of 70% being broadcast by that radio station;
whereas the radio station would broadcast R-3 with a probability of 5% even if there
were no police radar station. The responses of the motorists who listen to this
information include:
 5% of them would speed if R-3 is broadcast
 90% of them would speed if R-3 is NOT broadcast
Assume that
 All motorists listen to this radio station
 The existence of a radar station on R-3 and motorists who listen to this
ratio station are statistically independent
Determine:
d. the probability that R-3 is broadcast
e. the probability of speeding after the radio station is in operation
Problem 3 (30%)
Consider the cross section of a steel pipe of thickness t as shown below. Cracks may
develop from the inner towards the outer surface, and the crack length X is a random
variable that takes on continuous values between 0 and t.
x
crack
t
Based on a histogram plotted using a large amount of data on pipes of this type, an
engineer finds that only 20% of the cracks went beyond half of the pipe thickness, and
also that the PDF fX(x) decays very rapidly within the thickness of the pipe, in the
form

fX(x) = ce
x
x0
(0 < x  t)
where c and x0 are suitable constants to be determined.
a. What would be the units for c? Explain briefly.
b. Set up two equations (in the form of integrals) that can be solved to determine c
and x0 (in terms of t). DO NOT solve these equations.
Now, if you worked things out correctly in part (a), the solutions for c and x0 should
be
1
x0 = 0.361t
and
c=
0.361 t (1  e 1/ 0.361 )
Using these values, and considering a 25-mm-thick pipe,
c. what is the probability that a crack will penetrate more than 16 mm?
d. suppose a crack exists, but an ultrasonic detector has determined that the crack tip
is at least 10 mm below the outer surface of the pipe. What is the probability that
the crack length is less than 5 mm?