A pharmaceutical company proudly publishes results from a trial of its new test

Question

A pharmaceutical company proudly publishes results from a trial of its new test for a certain genetic disorder. The false negative rate is small: the test returns a negative result for only 10% of patients with the disorder. The false positive rate is also small: the test returns a positive result for only 15% of participants that do not have the disorder. Assume that 1% of the population has the disorder. Let’s see how good a test this will be and what a test result would mean to you as a patient. Calculate your answers to 2 significant digits.

(a) What is the probability of having the disorder if you have a negative test result? (Seeing your answer, how reassured should you be if you were the one that had a negative test result?)
(b) What is the probability of having the disorder if you have a positive test result? (Seeing your answer, how anxious should you be if you were the one that had a positive test result?)
(c) Repeat part (a) assuming that 20% of the population has the disorder.
(d) Repeat part (b) assuming that 20% of the population has the disorder.

Explanation / Answer

a. Probability that someone have disorder is 0.01 (1% )

The probability that someone have disorder and result is negative is 0.1 (10% false negative)   

so the   total probability of having the disorder if someone have a negative test result is 0.01*0.1=0.001 or 0.1%

b. Probability that someone have disorder is 0.01 (1%)

The probability that someone have disorder and result is positive is 0.85 (15% false positive)   

so the   total probability of having the disorder if someone have a positive test result is 0.01*0.85=0.0085 or 0.85%

c. Probability that someone have disorder is 0.2 (20%)

The probability that someone have disorder and result is positive is 0.1(10% false negative)   

so the   total probability of having the disorder if someone have a negative test result is 0.2*0.1=0.02 or 2%

d. Probability that someone have disorder is 0.2 (20%)

The probability that someone have disorder and result is positive is 0.85 (15% false positive)   

so the total probability of having the disorder if someone have a positive test result is 0.2*0.85=0.17 or 17%

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