PhonePe Interview Question | Probability
Question
A test has a true positive rate of 100% and false positive rate of 5%. There is a population with a 1/1000 rate of having the condition the test identifies. Considering a positive test, what is the probability of having that condition?
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Answers ( 3 )
Incidence rate = 1/1000 = 0.001
true positive rate = 1
false positive rate = 0.05
We have formula = (incidence rate × true positive rate) / {(True positive rate × incidence rate) + (False positive rate × (1 – incidence rate)
Substituting the above values : (0.001 × 1)/{(1 × 0.001) + (0.05 × (1 – 0.001)}
(0.001) / (0.001 + 0.04995)
= 0.001 / 0.05095
= 0.01963
=0.01963*100 =1.963%
nice explanation
The formal Bayesian approach:
Let A = person has the condition
Let B = positive test
We wish to calculate p(A| B), It is vitally important that you understand that this is the probability sought.
Using Bayes’s theorem, we can compute this probability.
p(A| B) = p(B| A)p(A)/(p(B| A)p(A) + p(B| not A)p(not A))
Of these probabilities, which do we know?
p(A) = .001 (This is given in the problem.
p(notA) = .999 (Why? You compute this from knowing that p(A) = .001.)
p(B| notA) = .05 This is given to you in the problem.
p(B| A) = 1.00 This, too, is given in the problem.
So: p(A| B) = 1.00 x .001/((.001 x 1) + (.05 x .999))
= .01998