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Say a test of 11 questions, and a person's score is from 0-11, and is categorized into level 1 (score 0-3), level 2 (score 4-7), and level 3 (score 8-11).

The priors of the three levels are $p_1, p_2, p_3$.

Now randomly draw one question from the person's test, and see the question is answered correctly, what is the posterior that the person is level 1, 2 and 3.

With Bayes rule, $p(level \ 1)=\frac{p(correct|level \ 1)p_1}{p(correct|level \ 1)p_1+p(correct|level \ 2)p_2+ p(correct|level \ 3)p_3}$.

I am confused because of the categorization of the score into levels, which makes the calculation of $p(correct|level \ 1)$ difficult or is it possible at all with the available information?

If I know the priors of score 0 to score 11, then I can calculate $p(correct|score \ 0)=0/11$, $p(correct|score \ 1)=1/11$ etc to calculate the posteriors of the scores at least. Is this correct?

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  • $\begingroup$ The difficulty is more about the likelihoods. Can you calculate $p(\text{correct} \mid \text{level} \ 3)$ etc.? $\endgroup$ Commented Aug 24 at 21:33
  • $\begingroup$ @Henry I agree that you want to calculate $P( correct | level 3)$, However, as stated, I claim you cannot. I suspect that part of the question is missing and as such, it cannot be answered. I am assuming the person is looking for a number as the answer. $\endgroup$ Commented Aug 25 at 2:54
  • $\begingroup$ Few points that might matter. If the sample had been 5 random questions and all had been answered correctly, the person could be level 2 or level 3. Is the probability of answering a question independent of answering other questions? With 1 correct answer, I would conclude that P(level 1) = P(level 2) = P(level 3) = 1/3 $\endgroup$ Commented Aug 26 at 11:46

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