Bayes Theorem: You are a research engineer examiningle emuent (i.e. final discha

Question

Bayes Theorem: You are a research engineer examiningle emuent (i.e. final discharge) from a wastewater treatment plant in a developing country. You have been asked to assess the health risks associated with inadequate treatment. Analysis of the efflucnt indicates the following percentages of microorganisms: (20 points) Protozoans Heterotopic Bacteria Helminthes 15% 70% 15% For contact with proto7oans, the chance of illness is 35% For heterotrophic bacteria, the chance of illness is 22% Finally, for viruses, the chance of illness is 8% If someone gets an intestinal illness from direct contact with this effluent, what is the likelihood that the illness was caused by the protozoans? 24TAM lype here to search

Explanation / Answer

Here we are given that P( protozoans ) = 0.15, P( heterotopic bacteria ) = 0.7 and P( helminthes ) = 0.15

Also we are given the conditional probabilities that:

P( illness | protozoans) = 0.35, P( illness | heterotopic bacteria ) = 0.22 and P( illness | helminthes ) = 0.08

Using the law of total probability, the probability of illness is computed as:

P( illness ) = P( illness | protozoans) P( protozoans ) + P( illness | heterotopic bacteria ) P( heterotopic bacteria ) + P( illness | helminthes ) P( helminthes )

P( illness ) = 0.35*0.15 + 0.22*0.7 + 0.08*0.15 = 0.2185

Now given that someone has got an illness, probability that the illness was caused by protozoans, using the bayes theorem we get:

P( protozoans | illness ) = P( illness | protozoans) P( protozoans ) / P( illness )

P( protozoans | illness ) = 0.35*0.15 / 0.2185 = 0.2403

Therefore 0.2403 is the required probability here.

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