Answer to Question #270440 in Statistics and Probability for Light

Question

The probability that a cancer malaria test will detect the disease in a person who has the disease is 0.9. The probability that a person who does not have malaria will give a positive reading on the test is 0.2 (i.e. the test will give a positive reading even though the person does not have the disease). If 0.2% 0f the population has malaria during a certain period, what is the probability that a person selected at random will give:

(i) Positive test result?

(ii) Have malaria given that he/she shows a positive result?

(iii) A false negative test result?

Answer 1

Let A - "person have malaria", B - "test is positive", then

(i) According to law of total probability

$P(B)=P(A)*P(B|A)+P(\neg A)*P(B|\neg A)=0.002*0.9+0.998*0.2=0.2014$

(ii) According to Bayesian formula

$P(A|B)={\frac {P(B|A)*P(A)} {P(B)}}={\frac {0.9*0.002} {0.2014}}=0.009$

(iii)

$P(\neg B|A)=1-P(B|A)=1-0.9=0.1$