Did the student know it, or get lucky?

Question

A student genuinely knows the answer to 80% of the topics on an exam. If they know a topic, they answer its multiple-choice question correctly for sure. If they don't know it, they guess among the 4 options and are right with probability $\tfrac14$ .

You see them answer one question correctly.

What is the probability they actually knew that topic?

Accepted Answer

Let K be "knew it" and C be "answered correctly." We want P(K \mid C). P(C\mid K) = 1,\quad P(K) = 0.8,\qquad P(C\mid 	ext{not }K) = 	frac14,\quad P(	ext{not }K) = 0.2. The total probability of a correct answer is P(C) = 0.8	imes 1 + 0.2	imes 	frac14 = 0.8 + 0.05 = 0.85. By Bayes, P(K\mid C) = \frac{0.8}{0.85} = \frac{16}{17} \approx 0.941. So a correct answer means about a 94% chance they truly knew it, the small guessing contribution barely dents it. Takeaway. A correct answer is weak evidence of knowledge only when guessing is easy. Here lucky guesses add just 0.05 to the 0.80 of gen…

Did the student know it, or get lucky?

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