If a person has a rare disease, then a diagnostic test for that disease detects

ID: 3209300 • Letter: I

Question

If a person has a rare disease, then a diagnostic test for that disease detects it with probability 99%. If a person does not have the disease, then the test reports that she does not have the disease with probability 95%. Only 0.5% of the population has the disease. a) If the test reports a positive result for a person selected at random from the population, find the probability that this person actually has the disease. b) To improve the value of probability in part a, which is better: reducing false positives by 20% or reducing false negatives by 20%? Explain your answer.

Explanation / Answer

Have disease (0.5%)

Do not have disease (99.5%)

Test Positive

0.99

0.05

Test Negative

0.01

0.95

P (True Positive) =0.99*0.005=0.00495

P (True negative) =0.95*0.995=0.94525

P (False Positive) =0.05*0.995=0.04975

P (False negative) =0.005*0.01=0.00005

0.00495/(0.00495+0.04975)

0.00495/0.0547

0.09

Reducing false positives by 20% would give a better probability because the value of denominator would decrease and consequently the value of probability would go up.