CIV 360/548: Risk Assessment and Management
Prof. Erik VanMarcke
PROBLEM 1
EXPLORATION DECISION ANALYSIS -- FOLLOW-UP TO PRECEPT #1
This problem is a follow-up to the "lab/precept" on searching for solution cavities in limestone areas. We consider herein two types of extensions: (a) the cavity sizes are random, and (b) the chance of structural failure ("event F") depends on the size of the cavity (provided there is one). Once a cavity is detected, the probability of structure failure is assumed to be zero (as preventive measures will then be taken). The following information is provided (Regarding notation: an "underbar" is used to indicate that what follows is a subscript; for instance, A_0 should be read as "A-subscript-zero". Otherwise, the notation is consistent with that used during Precept #1):
and
As during the precept, ignore the chance that there is more than one cavity and assume that a cavity, if there is one, is equally likely to be centered at any point on the site. The site is a square and cavities are spherical, measured by the ratio of their diameter d to the site dimension L. Small, medium-size, and large cavities have ratios d/L = 0.1, 0.2, and 0.4, respectively.
(a) Define relevant events, and plot an "event tree" and a "probability tree" for this uncertain situation (or "experiment").
(b) Find the probability of failure P[F] before any exploration is done.
(c) Find the probability that failure, should it occcur, will be caused by a small cavity. Then evaluate same for a "medium-size" and "large-size" cavity.
(d) A "simultaneous" search consists of making "n" borings and observing the overall outcome (i.e, cavity is either detected or not detected). If a cavity is found, it will be filled with grout, preventing failure. Estimate the "new" probability of failure based on a program of simultaneous search involving "n" borings. Plot this probability (approximately) as a function of n.
(e) Estimate the optimal number of borings in a simultaneous search program if the cost per borehole is $1,000 and the cost of failure is $250,000. Ignore the drill-rig "mobilization charge". Check the sensitivity of your result to the costs mentioned; specifically, how high or low would the "cost per borehole" have to be for the "optimal" number of borings to change?
(f) If two widely-spaced borings are made and a cavity is not found, estimate the "posterior" probabilities that it is small, medium-size, or large.
(g) If two widely-spaced borings are made and the cavity is not found, update the probability of failure.
PROBLEM 2
(Expressing a State of Knowledge about the Cause of the Crash of
Flight TWA 800)
Assume, as is widely thought, that there are only these two possible causes of the crash of TWA 800:
A = {a bomb or a missile downed the plane}
A' = {mechanical or structural failure occurred}
Investigators estimate the (prior) probability that event A' occurred at one in ten (or 0.1), before examining the wreckage of the plane. Examination of wreckage will result in one of these events:
B = {indication of a bomb or missile impact}
B' = {no such indication}
Either way, the true cause of the crash will remain uncertain, but there will be more information than before. The investigators also express the following conditional probabilities (related to what they expect the search of the wreckage will yield):
P[B|A] = 0.8
P[B'|A'] = 0.9
Questions:
(a) Draw an event tree and corresponding probability tree, and evaluate the probability, P[B], that the search will point to a bomb or a missile as the cause of the crash.
(b) Use Bayes' theorem to evaluate the "posterior probability" that a bomb or a missile downed the plane
(i) given that the search indicates this cause
(ii) given that the search fails to indicate this cause(c) Is your subjective judgment about the posterior probabilities mentioned in Part (b) consistent with the numerical values obtained using Bayes' theorem? In other words, are you "surprised" about your answers in Part (b), and if so, why?
(What sources of bias in "information processing" does this reveal, if any?)
(As an aid in answering (c), you might want to sketch your "subjective probability distribution" for P[A|B] and P[A|B'] based solely on the information provided in the problem statement.)
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