Question

For disease X, there’s a new rapid and accurate test in the market. The sensitivity of the test is 90% (probability to correctly detect disease X with positive test result), and the specificity of the test is 99% (probability to correctly give negative result for disease X free case). The prevalence of the disease is 0.01% in the population. Write in as percentage without percentage sign and with three decimal places. What is Pr[ X | positive ]? (i.e. probability that a person with positive test result has disease X?)

Answer 1

X : Event of a person has disease X

P(X) = 0.01/100 = 0.0001

: Event of a person not has disease X

P() = 1-P(X) = 1-0.0001 = 0.9999

P : Event of test result positive

N : Event of test result negative

Probability to correctly detect disease X with positive test result : P(P|X) = 90/100 = 0.9

Probability to correctly give negative result for disease X free case : P(N|) = 99/100 = 0.99

Probability to give positive result for Disease X free = P(P|) = 1-0.99 = 0.01

probability that a person with positive test result has disease X = P(X|P)

By Bayes theorem,

Probability that a person with positive test result has disease X = 0.008920607

Pr[ X | positive ] = 0.008920607 x 100 = 0.8920607

Pr[ X | positive ] = 0.892