Subject- Data Analysis and Decision Making 1. A doctor examines a patient and th
ID: 3064091 • Letter: S
Question
Subject- Data Analysis and Decision Making
1. A doctor examines a patient and thinks the patient might have a disease that is very rare -- 1 in 1 million. The patient goes for a test which is 99.9% accurate -- meaning false positives as well as false negatives occur only .1% of the time. a) If the test comes back positive, what is the probability that the patient has the disease? b) If, after seeing the test results, the patient decides to have another lab administer the same test and the test results come back positive again, what is the probability that he has the disease now?
Hint: You can use a Bayesian calculator such as: http://psych.fullerton.edu/mbirnbaum/bayes/BayesCalc.htm
Explanation / Answer
Pr(Disease) = 1/1000000 [Let say the event is A
so, Pr(A) = 1/1000000
Pr(A') = 999999/1000000
Pr(False positive) = Pr(False negative) = 0.1/100 = 0.001
so Here Pr(False positive) = 0.001
Let say Pr(test postivie) = B
Pr(Test positve when the person dosn't have disease) = Pr(B l A') = 0.001
Pr(False negative) = Pr(Test negative when the person have the disease) = Pr(B' l A) = 0.001
(a) Pr(Disease l Test comes postive) = Pr(A l B) = Pr(B l A) * Pr(A) / [Pr(B l A) * Pr(A) + Pr(B l A') * Pr(A')]
= 0.999 * (1/1000000)/ (0.999 * 1/1000000 + 0.001 * 999999/1000000) = 9.98/10000
(b) Now he went for the the another lab for the same test and the result comes positive again.
Pr(He has the disease) = 1 - Pr(Both test gives the wrong results) = 1 - (1 - 9.98/10000) * (1 - 9.98/10000) = 0.002