Here is a simple statistics question I was given. I'm not really sure I understand it.
X = the number of aquired points in an exam (multiple choice and a right answer is one point). Is X binomial distributed?
The professor's answer was:
Yes, because there is only right or wrong answers.
My answer:
No, because each question has a different "success-probability" p. As I did understand a binomial distribution is just a series of Bernoulli-experiments, which each have a simple outcome (success or failure) with a given success-probability p (and all are "identical" regarding p). E.g., Flipping a (fair) coin 100 times, this is 100 Bernoulli-experiments and all have p=0.5 . But here the questions have
different kinds of p right?
Best Answer
I would agree with your answer. Usually this kind of data would nowadays be modeled with some kind of Item Response Theory model. For example, if you used the Rasch model, then the binary answer $X_{ni}$ would be modeled as
$$ \Pr \{X_{ni}=1\} =\frac{e^{{\beta_n} - {\delta_i}}}{1 + e^{{\beta_n} - {\delta_i}}} $$
where $\beta_n$ can be thought as $n$-th persons ability and $\delta_i$ as $i$-th question difficulty. So the model enables you to catch the fact that different persons vary in abilities and questions vary in difficulty, and this is the simplest of the IRT models.
Your professors answer assumes that all questions have same probability of "success" and are independent, since binomial is a distribution of a sum of $n$ i.i.d. Bernoulli trials. It ignores the two kinds of dependencies described above.
As noticed in the comments, if you looked at the distribution of answers of a particular person (so you don't have to care about between-person variability), or answers of different people on the same item (so there is no between-item variability), then the distribution would be Poisson-binomial, i.e. the distribution of the sum of $n$ non-i.i.d. Bernoulli trials. The distribution could be approximated with binomial, or Poisson, but that's all. Otherwise you're making the i.i.d. assumption.
Even under "null" assumption about guessing, this assumes that there is no guessing patterns, so people do not differ in how they guess and items do not differ in how they are guessed--so the guessing is purely random.