dave2008
Legend
What we are saying is that the rules don't allow for us to set the DC at 15 for a trained individual and 20 for a non-trained individual for the same task. Nor do the rules allow for both the trained and non-trained individual to roll 15 on the die and get different results. When I do that I am engaging in a house rule.
How do the rules not allow you to do this? I just did a quick review of the PHB and DMG and I didn't see anything that says you can't do that. It doesn't fall into the guidelines they provide, but they are pretty clear that it is DM judgement and you are free to use different values. And it is not provided as an example on how to do things, but it is not labeled as something you can't do either. There are even examples of auto-success for trained (proficient) individuals and suggestions for modifying DC based on character level. So again, how do the rules "prevent" you from playing this way?
Also, the rules do allow a trained and untrained to get different results on a roll of 15. That is basically how the system works, the trained character adds the proficiency bonus which can provide different results based on the DC.