SPRING 2000

CEE 360/548: Risk Assessment and Management

Prof. E.VanMarcke

ASSIGNMENT #1

This problem is a follow-up to the "lab/precept" on searching for solution cavities in limestone areas. We consider two types of extensions: (a) cavity sizes are random, and (b) the chance of structural failure ("event F") depends on the size of the cavity (if 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 (Re notation: an "underbar" indicates a subscript; for instance, A_0 should be read as "A-subscript-zero". Otherwise, the notation is consistent with that used during Precept #1):

• P[A_0] = P[no cavity] = 0.3
• P[A_1] = P[small cavity] = 0.5
• P[A_2] = P[medium-size cavity] = 0.15
• P[A_3] = P[large cavity] = 0.05

and

• P[F|A_0] = 0
• P[F|A_1] = 0.1
• P[F|A_2] = 0.5
• P[F|A_3] = 1

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 ("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; in particular, 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.

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