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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 polic