[Bayes\' Rule] A given acute respiratory symptom can occur as a result of three

Question

[Bayes' Rule] A given acute respiratory symptom can occur as a result of three known conditions, as well as for unknown reasons. It occurs in 2% of colds, 35% of fungal pneumonia sufferers, and 70% of tuberculosis patients, as well as 0.001% of others (usually asa. resultof allergies). At given time, 4% of the population have colds, 0.02% have fungal pneumonia, and 0.005% have tuberculosis-and assume for simplicity that these groups are disjoint. a. Assume we have detected an instance of the symptom. What is the probability that we have found someone with fungal pneumonia? What is the probability that it is not a result of any of the three conditions? b.

Explanation / Answer

a)P(instance of acute respiratory system)=P(cold and  acute respiratory system)+P(fungal pneumonia and  acute respiratory system)+P(tuberculosis and  acute respiratory system)+P(others and  acute respiratory system)

=0.04*0.02+0.0002*0.035+0.00005*0.7+(1-0.04-0.0002-0.00005)*0.00001=0.000915

hence P(fungal pneumonia given acute respiratory system)

=P(fungal pneumonia and  acute respiratory system)/P(acute respiratory system)

=0.0002*0.035/0.000915=0.076536 ~ 7.6536%

b)

P(not a result of three conditions given acute respiratory system)

=P(others and  acute respiratory system)/P(acute respiratory system)

=(1-0.04-0.0002-0.00005)*0.00001/0.000915=0.010494~ 1.0494%

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