Thinking about it more I think it makes more sense to have N as the number of test takers. That way you can model it as a question x having a probability p_x of

Thinking about it more I think it makes more sense to have N as the number of test takers. That way you can model it as a question x having a probability p_x of being correct. Then the total grade on problem x will be a binomial(N, p_x), which will converge to a normal distribution for large number of test takers. Then you are simply summing up a small number of normal approximations to get the final grade (and sum of normals is normal). Did I model that correctly?