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The classic way to do that - which works well, again, in 5e - is to simply take the roll behind the screen. Another thing you could do back in the day is not tell the player whether he wanted high or low (since there were variants that went either way).
This is another thing I addressed in the first draft.
Beside just hating it when DMs roll for their players, rolling behind the screen doesn't really leave uncertainty. At most it can conflate negatives with false negatives. "You can't tell." But it can't produce believable false positives, not unless the DM improvises, or uses a house rule like "Natural 1 produces opposite answer." And the problem with THAT (in my mind) is that the chance of a false positive is always 1/20, regardless of skill or roll.
So I was looking for a system where the player gets to roll their own dice, and see their own dice, and the strength of the roll influences their faith in the answer...but always leaving a realistic and immersive seed of doubt.
If your players really like the feel of rolling dice themselves, an alternate little trick is to roll a d20 behind the screen as a 'seed.' You add it to the player's natural roll, subtracting 20 if the result is higher than 20 - thus you get the flat 1-20 range of a normal roll, but the player has no idea if he actually rolled 'well' or not.
Yes, I considered using this as the secondary roll (I'm fairly convinced that a complementary secret roll is necessary to achieve what I'm looking for) but I like how the probabilities work out with my system better.
The only thing my system does not provide is the possibility of convincing false negatives, which I admit is a shortcoming.
A system like you want could also address part of the 'DM may I' problem, at the price of adding another step to resolution. Players could ask the DM whether their PCs believe they can do something, make a check, and have some idea, but some remaining uncertainty as to whether they're able or not. :shrug:
Yes, absolutely.
"Can I disarm it?" (roll)
(secret roll) "Looks feasible."
"Ok, I try, do I succeed?" (roll)
(secret roll) "Seems like it."
"Ok, I open the chest."
...click...
BOOM!!!
By the way, the scenario I was really thinking about with all of this was pathfinding, and there I LOVE the ambiguity. You come to a junction in the cave system, and you roll some skill to determine the best way, and the DM gives you an answer, but you may never know for sure if it was in fact the best way. Now that's immersion.