Exercise 3.9. (SUSPENSION BRIDGES ARE GOOD ASSETS) A civil engineering company,

Question

Exercise 3.9. (SUSPENSION BRIDGES ARE GOOD ASSETS) A civil engineering company, based on past experience, believes that in 40% of cases bridges collapse due to construction problems, 45% of cases bridges collapse due to design problems, and 15% of cases bridges collapse due to other problems such as earth quakes, heavy vehicular loads etc. Danry is a good designer and claims that there is only log( DGC) % chance that his design may fail, and also Chris is a good constructor and claims that there is only 0.5log(DGCy . chance that his construction may collapse, and assume other 15% chances of collapsing remain the same. Fig. 3.9. Danny Chris Suspension Bridge Assume the construction company bought a design from Dannry and later hired Chris as constructor. Unfortunately, the suspension bridge collapsed within the warranty period. (a) What is the conditional probability that the bridge collapsed due to a design problem? (b )What is the conditional probability that the bridge collapsed due to a construction problem?

Explanation / Answer

Let C be the event that bridge collapse due to construction problems, D be the event that bridge collapse due to design problems and O be the event that bridge collapse due to construction problems.

Let B be the event that bridge collapse.
Let C be the event that construction may fail, D be the event that design may fail and O be the event that there are other problems.

Then P(B|C) = 0.4, P(B|D) = 0.45, P(B|O) = 0.15

log(DGC) = log(492) = 6.2
0.5log(DGC) = 0.5log(492) = 3.1

P(D) = 6.2%, and P(C) = 3.1%, P(O) = 15%

By law of total probability,
P(B) = P(B|C) P(C) + P(B|D) P(D) + P(B|O) P(O)
= 0.4 * 0.031 + 0.45 * 0.062 + 0.15 * 0.15 = 0.0628

(a)
Conditional probability that bridge collapsed due to a design problem is
P(D|B) = P(B|D) * P(D) / P(B) (By Bayes theorem)
= 0.45 * 0.062 / 0.0628 = 0.4443

(b)
Conditional probability that bridge collapsed due to a construction problem is
P(C|B) = P(B|C) * P(C) / P(B) (By Bayes theorem)
= 0.4 * 0.031 / 0.0628 = 0.1975

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