Question

Consider a small town that has a population of 1,000 people. It is known that in this town, 10 people are infected with a rare disease. The remaining 990 people are NOT infected with the disease. This data is known with certainty. Recently, The FDA (Food and Drug administration) developed a test that determines if a person is infected with this disease. However, as with most test of this nature, it is not foolproof proof as there are a certain percentage of false positives and false negatives that result from the test. In order to determine the effectiveness of this test, all of the towns people have agreed to participate in an experiment involving FDA testing to determine the disease. The results of the test are shown in the table below. Basically, of the 10 people who were known to be infected with the disease, 9 tested positive and one tested negative. Of the 990 people who were known to NOT be infected with the disease, 99 tested positive and 891 tested negative. Answer the following questions based on this information: (20 pts) a. What is the probability that a randomly selected person in the town is infected with the disease? b. What is the probability of thase who tested positive? c. Given a person is infected with the disease, what is the probability of testing positive d. Given a person tested positive, what is the probability they are infected with the disease? e. Are the events testing positive and being infected with the disease independent or dependent events? Prove why or why nat. InfectedI) Not Infected (NI Totals 108 892 1000 Test Positive (+) Test Negative 9 891 Total

Answer 1

a)  
P(infected with disease) = 10/1000=   0.01
  
b)  
P(tested positive) = 108/1000=   0.108
  
c)  
P(test positive| disease) = P(+ve test and disease)/P(disease) = 9/10=   0.9
  
d)  
P(disease| test +ve) = P(disease and +ve test)/P(+ve test) = 9/108=   0.083333333
  
e)  
two events to be independent  
P(A)P(B) must be equal to P(A and B)  
  
now, here  
P(postive test) =    0.108
P(disease) =    0.01
and P(positive and infected with disease) = 9/1000=   0.009
  
since, P(postive test)*P(infected disease)╪P(postive and disease)  
so, events are not independent