In the large N limit, and only the information that the mean is exactly 3.5, the obvious conclusion is that one is in a thought experiment, because that's an absurd thing to choose to measure and an adversary has chosen the result to make us regret the choice.

More generally, one should revisit the hypothesis that the rolls of the die are independent. Yes, rolling only 1 and 6 is more likely to get a mean of 3.5 than rolling all six numbers, but still quite unlikely. Model checking!