The proportion of people in a given community who have a certaindisease is 0.005
ID: 2918664 • Letter: T
Question
The proportion of people in a given community who have a certaindisease is 0.005. A test is available to diagnose thedisease. If a person has the disease, the probability thatthe test will produce a positive signal is 0.99. If a persondoes not have the disease, the probability that the test willproduce a positive signal is 0.01.
A) If a person tests negative, what is the probability that theperson actually has the disease?
B) For many medical tests, it is standard procedure to repeatthe test when a positive signal is given. If repeated testsare independent, what is the probability that a man will testpositive on two successive tests if he has the disease?
C) Assuming repeated tests are independent, what is theprobability that man tests positive on two successive tests if hedoes not have the disease?
D) If a man tests positive on two successive tests, what is theprobability that he has the disease?
Explanation / Answer
a) define the changes: P(disease) = 0.005 Conditional probabilities: P(Test +| Disease +) = 0.99 P(Test - | Disease + ) = 0.01 P(Test + | Disease -) = 0.01 P(Test - | Disease -) = 0.99 P(Disease)*P(Test - | Disease + )=0.005*0.01 =0.00005 b)Assume the person is sick, so he is one ofthe 200 persons with the disease P(Test +| Disease +) *P(Test +| Disease +) = 0.99*0.99= 0.98 c) P(Test + | Disease -) * P(Test + | Disease - )= 0.01*0.01=0.0001 d) P( a man tests positive on two successive tests and he has thedisease) = 1- P(two positive tests on two succesive tests and hedoes not have the disease) 1- P(no disease)*P(Test + | Disease -)*P(Test + | Disease -) = 1-0.95*0.01*0.01= 0.999905 Good Luck!
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