Very true! It depends on the particular variation[1] of the game being played. Which is one difficulty of MH - the ambiguity. Probabilities change when the initial conditions are slightly different or when the problem is interpreted differently. I've never assumed Monty picks at random. But it's entirely possible someone else would make that assumption, in turn changing the optimal strategy.

[1] https://en.wikipedia.org/wiki/Monty_Hall_problem#Variants