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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Dhruv2301 55 years 3 Answers 884 views Great Grand Master 0

Answers ( 3 )

  1. Sai
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    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%

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    Reply

    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

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