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I am doing an exercise question:

95% of the clients are being diagnosed correctly to have a disease, given that they have the disease.

90% of them being diagnosed correctly, given that they don't have the disease.

There are 0.025% of people have the disease. What is the probability that a person has the disease, given that the result is positive.

My work is:

Let A be event that whether the diagnose is correct

B be whether a person has the disease $$P(A=Yes|B=Yes)=0.95$$ $$P(A=Yes|B=N0)=0.9$$ $$P(B=Yes)=0.00025$$ $$P(A=No|B=Yes)=1-0.95=0.05$$ $$P(A=No|B=No)=1-0.9=0.1$$ And I have to find $P(B=Yes|A=Yes)$

That's all I can do. I think I have to find $P(A=Yes)$? May I ask how can I do so?

Ethan Bolker
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BigData
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    Related: http://math.stackexchange.com/questions/1308656/statistics-why-doesnt-the-probability-of-an-accurate-medical-test-equal-the-pr/1309733#1309733,http://math.stackexchange.com/questions/1497302/how-do-you-calculate-pab-if-you-know-only-pa-pb-neg-a/1497306#1497306 – Ethan Bolker Sep 09 '16 at 14:59
  • Also related: https://math.stackexchange.com/questions/32933/describing-bayesian-probability – Henry Oct 08 '20 at 09:36

1 Answers1

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May I ask how can I do so?

Certainly.

Use the Law of Total Probability:$$\mathsf P(X)~=~\mathsf P(X\mid Y)~\mathsf P(Y)+\mathsf P(X\mid Y^\complement)~\mathsf P(Y^\complement)$$

Then apply Bayes' Rule.

Graham Kemp
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