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.
What is the probability the test comes back positive?
What is the probability the test comes back positive and the person actually has the disease?
If the test comes back positive, what then is the conditional probability that he actually does have the disease?
Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website! 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.
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