Question

Suppose 10000 people are given a medical test for a disease. About1% of all people have this condition. The test results have a 15% false positive rate and a 10% false negative rate. What percent of the people who tested positive actually have the disease?

Answer 1

Answer:

The percent of the people who tested positive actually have the disease is 38.64%.

Step-by-step explanation:

Denote the events as follows:

X = a person has the disease

P = the test result is positive

N = the test result is negative

Given:

$P(X)=0.01\\P(P|X^{c})=0.15\\P(N|X)=0.10$

Compute the value of P (P|X) as follows:

$P(P|X)=1-P(P|X^{c})=1-0.15=0.85$

Compute the probability of a positive test result as follows:

$P(P)=P(P|X)P(X)+P(P|X^{c})P(X^{c})\\=(0.85\times0.10)+(0.15\times0.90)\\=0.22$

Compute the probability of a person having the disease given that he/she was tested positive as follows:

$P(X|P)=\frac{P(P|X)P(X)}{P(P)}=\frac{0.85\times0.10}{0.22} =0.3864$

The percentage of people having the disease given that he/she was tested positive is, 0.3864 × 100 = 38.64%.