"A certain disease has an incidence rate of 2%. If the false negative rate is 10% and the false positive rate is 1%, compute the probability that a person who tests positive actually has the disease."Why do we need to use Bayes' Theorem for this question?

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&quot;A certain disease has an incidence rate of 2%. If the false negative rate is 10% and the false positive rate is 1%, compute the probability that a person who tests positive actually has the disease.&quot;Why do we need to use Bayes&#039; Theorem for this question?

Superina0xb4i · Accepted Answer

A false positive is when the test is positive even though the person doesn&#039;t actually have the disease. The complement probability of a false positive is a true negative (i.e.: the test is negative and the person doesn&#039;t have the disease).- 1% (the false positive rate) represents the probability that the test is incorrect given that the person doesn&#039;t have the disease.- 99%  (the true negative rate) represents the probability that the test is correct given that the person doesn&#039;t have the disease.Neither of these two probabilities are what we want, since we don&#039;t know whether or not the person has the disease. All we know is that they tested positive (it could be a true positive or a false positive).In general, there are four possibilities:- True Positive: They had the disease (0.02) and the test was positive (0.9).- False Negative: They had the disease (0.02) but the test was negative (0.1).- False Positive: They didn&#039;t have the disease (0.98) but the test was positive (0.01).- True Negative: They didn&#039;t have the disease (0.98) and the test was negative (0.99).If a test is positive, then it could be a True Positive or a False Positive. We want to find the probability that, out of these two options, it was a True Positive. This yields:            Pr      [          True Positive      ]              Pr      [          True Positive or False Positive      ]        =            (      0.02      )      (      0.9      )              (      0.02      )      (      0.9      )      +      (      0.98      )      (      0.01      )        =      90    139    =  0.6474  …

"A certain disease has an incidence rate of 2%. If

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