A chemical engineer is interested in determining whether Mercury is present in a

Question

A chemical engineer is interested in determining whether Mercury is present in a fish product. The probability of Mercury being present is 4% (and thus the probability of its absence is 96%). The Acme Mercury Test as an 80% probability of detecting Mercury if it is present. The same tests has a 90% chance of not detecting Mercury if it is absent. Suppose the chemist selects a fish product at random and tests it for Mercury What is the probability that the product contains Mercury and that the contamination is detected by the test? Suppose a random product is tested and the test indicates that the product contains Mercury. What is the probability that the product does indeed contain Mercury?

Explanation / Answer

a. P(contains mercury and positive) = P(contains mercury) * P(positive | contains mercury)

= 4% * 80% = 0.032 = 3.2%

b. By theorem of total probability,

P(positive) =P(positive | contains mercury) * P(contains mercury) + P(positive | doesnt contain mercury) * P(doesnt contain mercury)

= 80% * 4% + 10% * 96% = 12.8%

By Bayes theorem,

P(contains mercury | positive) = P(positive | contains mercury) * P(contains mercury) / P(positive)

= 80% * 4% / 12.8% = 25%

= 0.25

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