Duestion 3 (10 pnts) Its learned from extensive computational models that 0.01%

Question

Duestion 3 (10 pnts) Its learned from extensive computational models that 0.01% of dams in the US will fail. (Assume that this has been or could be empirically validated). However, it's really important to know which dams are likely to fail. A UMD graduate student created an algorithm that reports "fail zone" or “no failure to aid decision-makers determine whether a given dam will fail. The algorithm isn't perfect. (Which algorithm is?) Of those dams in imminent threat of failure, 99.9% will reported in the “fail zone via this algorithm. Also, 25% of dams not in imminent threat of failure, will be reported in the "fail zone.” Now, you're your town's engineer and your town has a large dam. Your entire town happens to be downstream of the dam. Hearing about this algorithm, you decide to use it to see whether your town is under threat. It comes back positive, meaning “fail zone. What do you do? (ie, find the updated likelihood of failure, interpret, and briefly suggest a next step.)

Explanation / Answer

Let P(F) be the probability that dam is in imminent threat of failure (or will fail). Then P(F) = 0.01%

and P(~F) be the probability that dam is not in imminent threat of failure (or will not fail). Then P(~F) = 100% - 0.01% = 99.99%

Let P(FZ) be the probability that dam is reports that dam is in fail zone. Then,

Given that those dams in imminent threat of failure, 99.9% will reported in fail zone. So, P(FZ | F) = 99.9%

Given that those dams not in  imminent threat of failure, 25% will reported in fail zone. So, P(FZ | ~F) = 25%

By law of total probability,

P(FZ) = P(F) P(FZ | F) + P(~F) P(FZ | ~F)

= 0.01% * 99.9% + 99.99% * 25% = 0.25

We need to calculate the likelihood of failure, that is given that the algorithm suggests the dam in fail zone, the probability that the dam will fail. P(F | FZ)

By Bayes theorem,

P(F | FZ) = P(FZ | F) P(F) / P(FZ)

= 99.9% *  0.01% / 0.25

= 0.04 %

So, if the algorithm suggests the dam in fail zone, the likelihood of failure of dam is 0.04% which is very low. That is given that the algorithm suggests dam in fail zone, there is very little chance that the dam will fail  and the report of the algorithm can be discarded.

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