Question

c) Ujian diagnostik untuk penyakit mata adalah tepat 88%, dan 2.4% daripada penduduk sebenarmya mempunyai penyakit mata. A diagnostic test for eye disease is accurate 88%,and 2.4% of the population actually has eye disease i) Can kebarangkalian pesakit menguji positif. Find the probability the patient tests positive. (2 markah/marks) i) Cari kebarangkalian pesakit menguj negatf Find the probability the patient tests negative (2 markah/marks) ii) Cani kebarangkalian pesakit mempunyai penyakit mata, diberi dia menguj positif Find the probability the patient has eye disease, given he tests positive (3 markah/marks) v) Cari kebarangkalian pesakit tidak mempunyai penyakit mata, diberi dia menguji negatif Find the probability the patient does not have eye disease, given he tests negative (3 markah/marks)

Answer 1

Let E shows the event that person has eye disease. So we have

P(E) = 0.024

By the complement rule, the probability that person don't have eye disease is

P(E') =1 - P(E) = 1 - 0.024 = 0.976

Let P shows the event that person test positive and N shows the event that person test negative. So we have

P(P|E) = 0.88, P(N | E') = 0.88

By the complement rule, we have

P(N|E) = 1 - P(P|E) = 0.12

P(P|E') = 1- P(N|E') = 1 - 0.88 = 0.12

i)

The probability that patients test positive is

P(P) = P(P|E)P(E) + P(P|E')P(E') = 0.88 * 0.024 + 0.12 * 0.976 =  0.13824

ii)

The probability that patients test negative is

P(N) = P(N|E)P(E) + P(N|E')P(E') = 0.12 * 0.024 + 0.88 * 0.976 =  0.86176

iii)

P(E|P) = [ P(P|E)P(E) ] / P(P) = [ 0.88 *0.024] / 0.13824 = 0.1528

iv)

P(E' | N) = [P(N|E')P(E') ] / P(N) = (0.88 *0.976) / 0.86176 = 0.9967