Question

A disease affects 16% of the population. There is a test (not perfect) that detects disease with a probability of 98% (i.e comes back positive when the person has the disease). However, the test produces 5% false positives, i.e comes back positive even though the person does not have the disease.

i) A person who has the disease is tested, what is the probability that the test will come back negative.

ii) What is the probability that a randomly selected person tests positive for the disease?

iii) What is the probability that a randomly selected person who tested positive for the disease actually has the disease?

Answer 1

A disease affects 16% of the population

P(disease) = 0.16, P(no disease) = 0.84

test detects disease with a probability of 98%

Thus, P(detects positive and disease) = 0.16*0.98

i) A person who has the disease is tested, what is the probability that the test will come back negative.

= P(detects negative and disease) = 0.16*(1-0.98) = 0.0032

ii) What is the probability that a randomly selected person tests positive for the disease?

{the test produces 5% false positives, i.e comes back positive even though the person does not have the disease.

Thus, P(detects positive and no disease) = (1-0.16)*0.05 }

= P(detects positive and disease) + P(detects positive and no disease)

= 0.16*0.98 + (1-0.16)*0.05

= 0.1988

iii) What is the probability that a randomly selected person who tested positive for the disease actually has the disease?

= P(disease/ tested positive for the disease)

= P(disease and tested positive for the disease) / P(tested positive for the disease)

= 0.16*0.98 / 0.1988

= 0.7887