The test for a disease has a false positive rate of 5% and a false negative rate

Question

The test for a disease has a false positive rate of 5% and a false negative rate of 3%.  Suppose a person to be test has the disease with probability 20%.

If the test is positive, what is the revised probability that the person has the disease?

If the test is negative, what is the revised probability that the person has the disease?

If the test is negative, what is the revised probability that the person does not have the disease?

If the test is positive, what is the revised probability that the person does not have the disease?

Explanation / Answer

Ans:

Given that

P(positive/not disease)=0.05

P(negative/disease)=0.03

P(positive/disease)=1-0.03=0.97

P(disease)=0.20

P(not disease)=1-0.2=0.8

P(positive)=P(positive/disease)*P(disease)+P(positive/not disease)*P(not disease)

=0.97*0.20+0.05*0.80=0.234

P(negative)=1-0.234=0.766

1)P(disease/positive)=P(positive/disease)*P(disease)/P(positive)=0.97*0.20/0.234=0.829

2)P(disease/negative)=P(negative/disease)*P(disease)/P(negative)=0.03*0.20/0.766=0.008

3)P(not disease/negative)=P(negative/not disease)*P(not disease)/P(negative)

=(1-0.05)*0.8/0.766=0.992

4)P(not disease/positive)=P(positive/not disease)*P(not disease)/P(positive)

=0.05*0.8/0.234=0.171

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