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# hard probability

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A multiple choice exam has 4 choices for each question. A student has studied enough so that the probability they will know the answer to a question is 0.5, the probability that they will be able to eliminate one choice is 0.25, otherwise all 4 choices seem equally plausible. If they know the answer they will get the question right. If not, they have to guess from the 3 or 4 choices. As the teacher, you want to test to measure what the student knows. If the student answers a question correctly, what is the probability they knew the answer?

Feb 14, 2020

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$$P[\text{knew answer|correct answer}] = \\ \dfrac{P[\text{correct answer | knew answer}]P[\text{knew answer}]}{P[\text{correct answer}]}$$

$$P[\text{knew answer}] = \dfrac 1 2\\ P[\text{correct answer|knew answer}] = 1\\ P[\text{correct answer}] = \dfrac 1 2 + \dfrac 1 2 \left(\dfrac 1 4 \dfrac 1 3 + \dfrac 3 4 \dfrac 1 4\right) =\dfrac{61}{96}$$

$$P[\text{knew answer|correct answer}] = \dfrac{1 \cdot \dfrac 1 2}{\dfrac{61}{96}} = \dfrac{48}{61}$$

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Feb 16, 2020