Problem

For \(10,000\) cases with \(1\%\) prevalence, \(90\%\) sensitivity, and \(5\%\) false-positive rate, compute condition and noncondition counts, true and false positives, total positives, and \(P(\text{condition}~|~\text{positive})\).

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What this problem is really about

Base-rate problems become clearer when percentages are converted to expected counts in the stated population. Split the population by condition status first, then apply sensitivity only to the condition branch and the false-positive rate only to the noncondition branch. A positive-result probability must use all positives in its denominator, including both true and false positives.

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Use sensitivity 90% as posterior; with 90 true and 495 false positives, ignore the total-positive denominator 585.

There are 90 true positives and 495 false positives, for 585 positives; P(condition|positive)=90/585=2/13.

Use 100 false positives instead of 5% of 9900; combine with 90 true positives for 190 and report posterior 90/190.

Use 900 true positives from 1000 instead of 100 condition cases, plus 495 false positives, for 1395 positives and posterior 900/1395.

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For \(10,000\) cases with \(1\%\) prevalence, \(90\%\) sensitivity, and \(5\%\) false-positive rate, compute condition and noncondition counts, true and false positives, total positives, and \(P(\text{condition}~|~\text{positive})\).

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