Question 1206335: Suppose that a certain disease is present in 10% of the population, and that there is a screening test designed to detect this disease if present. The test does not always work perfectly. Sometimes the test is negative when the disease is present, and sometimes it is positive when the disease is absent. The following table shows the proportion of times that the test produces various results.
Test Is Positive
(P)
Test Is Negative
(N)
Disease
Present
(D)
0.07 0.03
Disease
Absent
(DC)
0.02 0.88
(a)
Find the following probabilities from the table. (Round your answers to two decimal places.)
P(D)=
P(DC)=
P(N|DC)=
P(N|D)=
(b)
Use Bayes' Rule and the results of part (a) to find
P(D|N).
(Round your answer to three decimal places.)
P(D|N) =
(c)
Use the definition of conditional probability to find
P(D|N).
(Your answer should be the same as the answer to part (b). Round your answer to three decimal places.)
P(D|N) =
(d)
Find the probability of a false positive, that the test is positive, given that the person is disease-free. (Round your answer to three decimal places.)
(e)
Find the probability of a false negative, that the test is negative, given that the person has the disease.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the following table applies:
P N
D .07 .03
DC .02 .88
P means test is positive
N means test is negative
D means disease is present
DC means disease is not present
(a)
Find the following probabilities from the table. (Round your answers to two decimal places.)
P(D)= .07 + .03 = .1
P(DC)= .02 + .88 = .9
P(N|DC)= .88
P(N|D)= .03
(b)
Use Bayes' Rule and the results of part (a) to find P(D|N).
(Round your answer to three decimal places.)
P(D|N) = p(N|D) * p(D) / p(N) = .03 * .1 / .9 = .0033333333 = .003.
(c)
Use the definition of conditional probability to find P(D|N).
(Your answer should be the same as the answer to part (b). Round your answer to three decimal places.)
P(D|N) = = p(D intersect N) / p(N)
from p(N|D), we can solve for p(D intersect N) as follows:
p(N|D) = p(N intersect D) / p(D) becomes .03 = p(N intersect D) / .1
solve for p(N intersect D) to get p(N intersect D) = .03 * .1 = .003.
p(N intersect D) is the same as p(D intersect N) = .003.
p(D|N) = p(D intersect N) / p(N) = .003 / .9 = .0033333333 = .003.
answer in part b and part c are the same, as they should be.
(d)
Find the probability of a false positive, that the test is positive, given that the person is disease-free. (Round your answer to three decimal places.)
probability of a false positive is p(P | DC).
that's the probability you will test positive given that you don't have the disease.
from the table, p(P | DC)) = .02.
(e)
Find the probability of a false negative, that the test is negative, given that the person has the disease.
probability of a false negative is p(N | D).
that's the probability you will test negative given that you have the disease.
from the table, p(N | D) = .03.
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