Question

The disease D has two symptoms, pain and fever. Pain occurs in 95% of the people with D, but also in 10% of the people without D. Fever occurs in 90% of the people with D, but also in 5 % of the people without D. D affects 1% of people. Which of pain or fever is a better indicator of D?

Answer 1

P[D] be the probability that person has disease D, P[D] = 1% = 0.01

P[D"] be the probability that person does not have disease D, P[D] = 1 - 0.01 = 0.99

P[ Pain | D ] be the probability that person has pain given he has disease D, P[ Pain | D ] = 95% = 0.95

P[ Pain | D" ] be the probability that person has pain given he does not have disease D, P[ Pain | D" ] = 10% = 0.10

P[ Pain ] be the probability that person has pain = P[ Pain | D ]*P[D] + P[ Pain | D" ]*P[D"] = 0.95*0.01 + 0.10*0.99 = 0.0095 + 0.0990 = 0.1085

P[ Pain, D ] be the probability that person has pain and has disease D = P[ Pain | D ]*P[D] = 0.95*0.01 = 0.0095

P[ Fever | D ] be the probability that person has fever given he has disease D, P[ Fever | D ] = 90% = 0.90

P[ Fever | D" ] be the probability that person has fever given he does not have disease D, P[ Fever | D" ] = 5% = 0.05

P[ Fever ] be the probability that person has fever= P[ Fever | D ]*P[D] + P[ Fever | D" ]*P[D"] = 0.9*0.01 + 0.05*0.99 = 0.009 + 0.0495 = 0.0585

P[ Fever, D ] be the probability that person has fever and has disease D = P[ Fever | D ]*P[D] = 0.9*0.01 = 0.009

To know , which is the better indicator we will calculate,

P[ D | Fever ] : he has disease D given he has fever = P[ Fever, D ]/P[ Fever ] = 0.009/0.0585 = 0.1538

and P[ D | Pain ] : he has disease D given he has pain = P[ Pain, D ] /P[ Pain ] = 0.0095/0.1085 = 0.0876

Since, P[ D | Fever ] >  P[ D | Pain ]

therefore fever is a better indicator of disease D