Consider the question of whether John Doe did well on his mathematics examination. This would seem like a straightforward thing depending on the questions and his answers. We can assume they are provided as part of the problem statement. We could also assume that a definition for “did well” is included. We could then consider a situation where under chaos theory, his act of taking the examination caused a hurricane that destroyed his answer sheet before it had been graded. This situation was not mentioned as either a possibility or non-possibility. However, we had the insight to consider it. Thus, we can say we don’t know if he did well on his mathematics examination, even though there is a straightforward answer.

Another possibility is that game show could have rigged things without telling us, with a 90% chance of the prize is behind behind door #1, a 9% chance that the prize is behind door #2, and a 1% chance that the prize is behind door #3. Which door was the initial choice would then decide whether the player should change the choice, rather than anything the host does. However, this was not told to us, but to avoid saying that choosing the other door is always the answer, we decide to question the uniformity of the probability distribution, despite there being no reason to think it is non-uniform. Thus, assuming that the game show might have altered the probability distribution, we can say not only that the host’s intent does not matter, but we don’t know the answer to the question.

To be clear, my counterpoint is that these considerations produce different problems and thus are not relevant.