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Setup: We consider a multiple choice test with 30 questions, each consisting of four potential answers. We know that (on average) neighbors answer 50% of all questions jointly correct.

Assumption: Let us assume that students pick randomly answers if they do not know the solution.

Question: How can we calculate the (expected) average share of identical incorrect answers?

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    $\begingroup$ "We know that (on average) neighbors answer 50% of all questions jointly correct." — I'm not sure what this means. Does it mean "for any question, any two students have a 50% chance of both answering the question correctly"? $\endgroup$ Dec 15 '17 at 18:16
  • $\begingroup$ Sorry for being unspecific. Exactly, this is what I had in mind. If we draw two random students from our sample, then the probability that both students gave the same answer is 50%. $\endgroup$
    – bachelor
    Dec 15 '17 at 18:19
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I interpret the main question to mean:

Among questions that have been answered incorrectly by both of two students, what is the expected proportion on which both students choose the same (incorrect) option?

By assumption, a student who doesn't know the answer chooses an option randomly (with equal probability of the four choices). Since we know both students answered the question wrong, we know that each chose one of the three incorrect choices. So there are 3 • 3 = 9 possible pairs of answers, 3 of which match. So $\tfrac{3}{9} = \tfrac{1}{3}$ is the chance of any one pair of incorrect responses matching. It follows that this is the expected proportion among all such questions.

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