Question 189051
In a region 4% of the population is thought to have a certain disease. A standard diagnostic test will correctly identify 92% of the people who have the disease. However, the test also incorrectly diagnoses 10% of those who do not have the disease as having the disease. A randomly selected person in the region is tested for the disease.
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Use a tree diagram with d,d' for diseased and not diseased
Use p,n for diagnosed positive and diagnosed negative
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What is the probability the test comes back positive?
P(p) = P(p and d) + P(p and d') = P(p}d)P(d) + P(p]d')P(d') 
= 0.92*0.04 + 0.1*0.96 = 0.1328
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What is the probability the test comes back positive and the person actually has the disease?
P(p and d) = P(p|d)*P(d) = 0.92*0.04 = 0.0368
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If the test comes back positive, what then is the conditional probability that he actually does have the disease?
P(d}p) = P(d and p)/P(p) = 0.0368/[P(p and d) + P(p and d')
= 0.0368/0.1328 = 0.2771
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Cheers,
Stan H.