It's very confusing. In particular it does not say that the box have 3 doors until the middle of the explanations. Also, I don't find the example very similar to the Bell's Inequality.

Moreover, I expect in a quantum system that when both open the same door they get the same result (or the oposite) so in a quantum system I expect that when both open the same door they get 100% (or 0%) agreement, so insted of 50% I expect 1/3 * 100% + 2/3 * 50 % = 66% (or 1/3 * 0% + 2/3 * 50 % = 33%).

Anyway, in some versions of the Bell's Inequality the doors of the boxes are "misalignment" so on box has white-gray-black doors and the other has another ser of colors. let's say creme-pink-brown doors. You never have a 100% or 0% of coincidences of the results.

[[EDIT - actually I take that back. The GHZ inequality refers to three systems whereas your link refers to three measurement choices]].

I don't think your link gives a derivation of the quantum correlations beyond "Quantum physics says that half the time they should get a match".

[0] https://en.wikipedia.org/wiki/Bell%27s_theorem#GHZ%E2%80%93M...