Question

You are the site engineer in charge of the operation of a set micro gas turbines in a factory. The correct operation of the turbines is an important factor in maintaining the energy supply for the machinery. Past experience indicates that there is a 98% chance that a turbine will operate normally in any particular day.

To ensure that a faulty turbine is identified and replaced, you order a test that measures the efficiency of each turbine and use that measure as an indication of possible problems in the turbine.
The test method is not perfect: the probability that a normally operating turbine will pass the test is 0.97 and the probability that a faulty turbine passes the test is 0.01.
The outcomes of the tests are statistically independent.
Suppose that one of the turbines fails the test, what is the probability of that turbine being faulty?

Answer 1

about 0.40

Explanation:

You have ...

• probability of operating normally = 0.98
• probability a normally operating turbine passes the test = 0.97
• probability a faulty turbine passes the test = 0.01

and you want to know the probability a turbine is faulty if it fails the test.

Test failure

The probability a turbine will fail the test is the sum of the probabilities a good turbine will fail, and that a bad turbine will fail.

p(good & fail) = p(good)×p(good fails) = 0.98×(1 -0.97) = 0.0294

p(bad & fail) = p(bad fails)×p(bad) = (1 -0.01)×(1 -0.98) = 0.0198

Then the probability a tested turbine will fail is ...

p(fail) = p(good & fail) +p(bad & fail) = 0.0294 +0.0198 = 0.0492

Conditional probability

The probability that a turbine is faulty given that it failed the test is ...

p(bad | failed) = p(bad & fail)/p(fail) = 0.0198/0.0492 ≈ 0.4024

The probability a turbine is faulty if it fails the test is about 0.40.

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