Devising a grading scheme that satisfies two tests

Question

Suppose two students take a $10$-question-long True/False quiz. One point is awarded for each correct answer.

Student $1$'s ten answers are given by $(F, T, F, T, T, T, F, T, T, F)$, where $F$ denotes false and $T$ denotes true. He receives a grade of $2/3$ (as a percentage, that is, $66.66\%$).

Student $2$'s ten answers are given by $(T, T, F, T, F, F, T, F, F, T)$. They receive a score of $1/6$ (as a percentage, that is, $0.16666\%$).

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What's a possible grading scheme that allows this to happen? Obviously, there has to be some sort of penalty for wrong answers, or maybe a penalty for wrong true answers only (and no penalty on wrong false answers)? I'm pretty sure the penalty will be $-1/3$ too, just from messing around.

I can't fit any grading scheme to satisfy this. Can someone please help me come up with one?

Answer 1

If student A got one wrong, the wrong answer would have to score $-8/3$. But in that case student B can have at most four right (they disagree on seven), and so would score negative.

If student A got two wrong, the wrong answers count $-2/3$ each. This is consistent with student B getting five right and five wrong, for an overall score of $5/3$. For example, this would be the case if the correct answers were F,T,F,T,T,T,F,T,F,T.

Answer 2

As a linear system, we do not have enough information to obtain a unique answer**. Let $x_i$ be the number correct for student $i$. Then assuming a penalty $p$ for incorrect answers, (and a linear penalty relationship,) the knowledge we have is $$x_1 - p(10-x_1) = 1/6 \\x_2 - p(10-x_2) = 4/6$$The goal is to solve for $p$, but we have two equations and three unknowns.

The other piece of information we have is that the second through fourth answers are the same between both students, and all other questions are opposites. Maybe someone can derive a relationship from that, finding combinations where the coefficients evenly divide the numerator(s)?

**If this is a homework problem, it should be sufficient to determine one of several possible models; however, if a student is considering arguing a grade with the professor, and maybe missed the review session where the scoring was discussed but is too embarrassed to admit that, it's important that we find one grading rubric. :-)