An Internet review of home pregnancy tests reports: "Home pregnancy testing kits usually claim accuracy of over 95% (whatever that may mean). The reality is that the literature contains information on only four kits evaluated as they are intended to be used-by women testing their own urine. The results we have suggest that for every four women who use such a test and are pregnant, one will get a negative test result. It also suggests that for every four women who are not pregnant, one will have a positive test result."15 For the purpose of this exercise, assume that the website's claims are correct (one in four pregnant women tests negative, one in four nonpregnant women tests positive).
a. Assume that 50% of all women who use these kits are actually pregnant. Use probability notation to denote and identify the probability of being pregnant, the probability of not being pregnant, the probability of testing positive if a woman is pregnant, and the probability of testing positive if a woman is not pregnant. Then find the overall probability of testing positive, and the probability of actually being pregnant, given that a woman has tested positive
b. Assume that 80% of all women who use the kits are pregnant. Find the probability of being pregnant, given that a woman has tested positive.
c. Now assume that only 20% of all women who use the kits are actually pregnant. Find the probability of being pregnant, given that a woman has tested positive.
d. Discuss how the above results suggest that, if a person tests positive for a disease that is quite rare, there tends to be just a small probability of actually having the disease.