Question

Suppose that one person in 10,000 people has a rare genetic disease. There is an excellent test for the disease; 98.8% of the people with the disease test positive and only 0.4% of the people who don't have it test positive.

A) What is the probability that someone who tests positive has the disease?
B) What is the probability that someone who tests negative does not have the disease?

Answer 1

A)The probability that someone who tests positive has the disease is 0.9995

B)The probability that someone who tests negative does not have the disease is 0.99999

Explanation:

Let D be the event that a person has a disease

Let
`D^c` be the event that a person don't have a disease

Let A be the event that a person is tested positive for that disease.

P(D|A) = Probability that someone has a disease given that he tests positive.

We are given that There is an excellent test for the disease; 98.8% of the people with the disease test positive

So, P(A|D)=probability that a person is tested positive given he has a disease = 0.988

We are also given that one person in 10,000 people has a rare genetic disease.

So,
`P(D)=(1)/(10000)`

Only 0.4% of the people who don't have it test positive.

`P(A|D^c)` = probability that a person is tested positive given he don't have a disease = 0.004

`P(D^c)=1-(1)/(10000)`

Formula:
`P(D|A)=(P(A|D)P(D))/(P(A|D)P(D^c)+P(A|D^c)P(D^c))`

`P(D|A)=(0.988 * (1)/(10000))/(0.988 * (1-(1)/(10000)))+0.004 * (1-(1)/(10000)))`

P(D|A)=
`(2470)/(2471)`=0.9995

P(D|A)=
`0.9995`

A)The probability that someone who tests positive has the disease is 0.9995

(B)

`P(D^c|A^c)`=probability that someone does not have disease given that he tests negative

`P(A^c|D^c)`=probability that a person tests negative given that he does not have disease =1-0.004

=0.996

`P(A^c|D)`=probability that a person tests negative given that he has a disease =1-0.988=0.012

Formula:
`P(D^c|A^c)=(P(A^c|D^c)P(D^c))/(P(A^c|D^c)P(D^c)+P(A^c|D)P(D))`

`P(D^c|A^c)=(0.996 * (1-(1)/(10000)))/(0.996 * (1-(1)/(10000))+0.012 * (1)/(1000))`

`P(D^c|A^c)=0.99999`

B)The probability that someone who tests negative does not have the disease is 0.99999