Question

This is a statistics problem.

A laboratory test for the detection of a certain disease give a positive result 5 percent of the time for people who do not have the disease. The test gives a negative result 0.3 percent of the time for people who have the disease. Large-scale studies have shown that the disease occurs in about 2 percent of the population.

What is the probability that a person selected at random would test positive for this disease?
For this one, I got 0.0194, so 1.94% but I wasn’t sure if the answer is correct.

What is the probability that a person selected at random who test positive for the disease does not have the disease?
This one was confusing to me… I got 0.71637, or 71.637% but I doubt this is right. Please help!

Answer 1

Consider the four possibilities and their probabilities.
has disease and tests positive:
(0.02)x(0.997)= 0.0199
has disease but tests negative:
(0.02)x(0.003)= 0.0001
has no disease and tests negative:
(0.98)x(0.95) = 0.9310
has no disease but tests positive:
(0.98)x(0.05) = 0.0490
Note the probabilities they add up to 1.000, as they should.
The answer to the first question (positive test probability) is
0.0199 + 0.0490 = 0.0689

The answer to the second question is:
0.0490/(0.0199+0.0490)= 0.7112

This is a rather large ratio of "false positives" and comes about because the disease is rare (2% of population) and there are many more false positives than "true" positives

Answer 2

That makes perfect sense. Thank you very much.

Answer 3

hey, we just did this problem in class today...this is from the ap stats 1197 free response prep...the first question's ans. is 0.06894 and the second ques.'s ans. is 0.710763