Bayes' Theorem and Bayesian Thinking

An interviewer asks, “A condition affects 1% of items. A test correctly identifies 90% of affected items, while 10% of unaffected items also test positive. Given a positive result, what is the probability that the item has the condition?”

Let$C$mean an item has the condition and$+$mean the test is positive. The target is:

The prompt does not ask for $P(+\mid C)$, even though the 90% detection rate gives us that quantity. This distinction is the high-risk part of the question. A test’s ability to detect known cases does not directly tell us how often a positive test identifies a real case.

A weak answer says 90%. That answer silently replaces the requested probability $P(C\mid +)$ with the supplied probability $P(+\mid C)$. The conditional bar changes the population in the denominator. For $P(+\mid C)$ ...