Question

In: Statistics and Probability

Assume there is a medical test to diagnose a disease. If a person has the disease,...

Assume there is a medical test to diagnose a disease. If a person has the disease, the probability of having positive test result is 98 percent. If a person does not have the disease, the probability of having negative test results is 99.6 percent. The probability that a person has a disease is 1 percent in the population.

Answer the following questions:

a) If a person has a positive test result, what is the probability that he/she has the disease?

b) If a person has a positive test result, what is the probability that s/he doesn’t have the disease?

c) If a person has a negative test result, what is the probability that he/she doesn’t have the disease?

d) If a person has a negative test result, what is the probability that s/he has the disease?

Note: Use the following notation in your answer:

D: Person with disease

ND: Person without disease

+T: Positive test result

- T: Negative test result

Write each question in the form of mathematical notation for conditional probability.

Calculate the answer using two methods:

1. Bayes’ rule and conditional probability equations.

2. Draw a table, assume a population (e.g. 1 million) and provide numerical answers

Expert Solution

Has the disease Does not have the disease Total

Test positive 9800 3960 13760

Test negative 200 986040 986240

Total 10000 990000 1000000

a) P(has the disease | Test positive) = P(has the disease and test positive )/P(Test Positive)

= 9800/13760 = 0.7122

b) P(Doesn't have the disease | Test positive) = P(Doesn't have the disease and test positive)/P(Test positive) = 3960/13760 = 0.2878

c) P(Doesn't have the disease | Test negative) = P(Doesn't have the disease and test negative)/P(Test negative) = 986040/986240 = 0.9998

d) P(has the disease | Test negative) = P(has the disease and test negative)/P(Test negative)

= 200/986240 = 0.0002

orchestra answered 1 year ago

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