ENGI 3423 Probability and Statistics

Faculty of Engineering and Applied Science
2010 Fall

Bayes Decision Tree Question

Aggregates for a highway pavement are extracted from a gravel pit.   Based on experience with the material from this area, it is estimated that the probabilities are

        P[G] = Pr(good-quality aggregate) = .70
        P[~G] = Pr(poor-quality aggregate) = .30

In order to improve this prior information, the engineer considers a testing programme to be conducted on a sample of the aggregate.   The test method is not perfectly reliable   –   the probability that a perfectly good-quality aggregate will pass the test is 80%, whereas the probability that a poor-quality aggregate will pass the test is 10%.

1. Draw both Bayes calculation and decision trees for the problem.   The decision tree should include the options “do not implement test programme” and “implement test programme”, as well as “accept the aggregate” and “reject the aggregate”.   Determine the optimal decision on the basis of maximum-expected-cost, given the following costs (on a nominal scale).   Accept, good aggregate, 15 units; accept, poor aggregate, –8 units; reject, good or poor aggregate, 5 units; cost of testing, 2 units.

2. If a sample is tested, and if it passes, what is the probability of good-quality aggregate?

3. If a second test is performed, and the sample again passes, what is the probability of good-quality aggregate?   One can use the result of part (a) as a prior probability in this case.