Chapter 4.5, Problem 15E

Screening Tests Suppose that a certain diseaseis present in 10% of the population, and that there is ascreening test designed to detect this disease if present.The test does not always work perfectly. Sometimesthe test is negative when the disease is present, andsometimes it is positive when the disease is absent. Thefollowing table shows the proportion of times that thetest produces various results.

a. Find the following probabilities from the table: $P\left(D\right),P\left(D^c\right),P\left(N|D^c\right),P\left(N|D\right)$ .
b. Use Bayes’ Rule and the results of part a to find $P\left(D|N\right)$ .

c. Use the definition of conditional probability to find $P\left(D|N\right)$ . (Your answer should be the same as theanswer to part b.)
d. Find the probability of a false positive, thatthe test is positive, given that the person isdisease-free.
e. Find the probability of a false negative, that thetest is negative, given that the person has thedisease.
f. Are either of the probabilities in parts d or e largeenough that you would be concerned about the reliabilityof this screening method? Explain.