Question

Jane is being tested for a rare disease. Only 1.5% of the population has the disease. The following data were collected. First a group of people who had the disease were tested. 98.5% tested positive and 1.5% tested negative. Next, a group of people who did not have the disease were tested. 3% tested positive and 97% tested negative. (First write down the six given probabilities) [1] Page 4 of 8 MATH 140 - Introduction to Statistics What is the probability that Jane will test positive for the disease? b) [2] Jane is tested for the disease and her test shows positive. What is the probability that Jane has the disease?

Answer 1

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Given,
Probability that a person will have a disease in the population is given by P	P(D)	0.15
Probability that a person tested positive given that he has disease is	P(+/D)	0.985
Therefore probability that person tested negative given that he has disease	P(-/D)
	1 - P(+/D)	0.015
Probability that a person tested positive given that he does not has disease is	P(+/ND)	0.03
Probability that a person tested negative given that he does not has disease is	P(-/ND)
	1 - P(+/ND)	0.97
Probability Jan will be tested positive for disease is given by	P(+)
Now, Jane can be tested positive either she is having a disease or not due to testing error probabilities
Probability that she would have disease is P(D)	0.15
Probability that she tested positive is P(+/D)	0.985
Probability that she would not have a disease is P(ND)	0.85
Probability that she tested positive is P(+/ND)	0.03
Therefore P(+) is given by	P(D)*P(+/D) + P(ND)*P(+/ND)
Value is	0.17325
Probability that Jane has disease given that she tested positive is given by	P(D/+)
We will apply bayes rule which is given as
P(A/B) = P(B/A) *P(A)/P(B)
Bayes gives simple reationship between probability of hypothesis before getting evidence and probability of hypothesis after getting evidence.
i.e. Relationship between probability of disease P(D) before tests were available to Probability of disease after tests were available.
Thus, P(D/+) = [P(+/D)*P(D)/P(+)]
P(+/D)	0.985
P(D)	0.15
P(+)	0.17325
Solution is	0.852813853