Question

Problem 1 [Sans R (a). Say a test can detect a disease with a type I error rate (false positive) of 10 % and a type II error rate (missed positive) of 0.1 %. If a person is randomly chosen from the population, the chance of having this disease is 0.1 %. If a random person is chosen from the population and tests positive for this disease, what is the probability they have this disease? (b). Say a test can detect a disease with a type I error rate (false positive) of 0.1 % and a type II error rate (missed positive) of 10 %. If a person is randomly chosen from the population, the chance of having this disease is 0.1 %. If a random person is chosen from the population and tests positive for this disease, what is the probability they have this disease?

Answer 1

a)

A - have the disease

B - test positive

P(A) = 0.001

P(B'|A) = 0.001

$\frac{P(B' \cap A)}{P(A)} = 0.001$

$\Rightarrow P(B' \cap A) = 0.001 P(A)$

$\Rightarrow P(A) - P(A \cap B)= 0.001 P(A)$

P(A ) - P(B and A) = 0.001 * P(A)

P(A and B) = 0.999 P(A) = 0.999*0.001

similarly

P(B|A') = 0.1 = P(B and A') /P(A') = 0.1

P(B) - P(A and B) = 0.1 * (1 - P(A))

P(B) - P(A and B) = 0.1*0.999 = 0.0999

P(B) = 0.0999 + P(A and B)

= 0.0999 + 0.999*0.001

= 0.100899

P(A|B) = P(A and B)/P(B) = 0.999*0.001 / 0.100899

= 0.00990099009

solve part b) similarly