The math is correct in your examples. However, that doesn't mean you were plugging in the right numbers.
LOL! Sorry I didn't give you my "Dissertation on the Comparative Distributions of Skill Checks Systems: An Examination of the Merits and Flaws of 2d10 versus d20." My point was to simply present a couple examples expounding the differences between the two for a moderate DC 10 and a difficult DC 20. Which was exactly what I accomplished.
Actually, these two facts together - the DC and modifier - leads into the most important point. That 2d10 and d20 do not have the same distribution over their length.
Of course they don't, that is obvious.
Between DC and modifier you determine what you need to roll (or better) on the dice.
I'm going to the rest because it really focuses on only the DC, when my entire point was about why it was a lousy example was about both in tandem. It's an incomplete look at what I was saying that makes so sense without the other half to determine what the roll is. If you thought that was all I was talking about then my apologies for not communicating clearly.
So you want to example the range of DC's over the range of likely applicable modifiers? That's fine, but that wasn't my intent as I felt such an over-analysis was unwarranted. I misunderstood your comment earlier, so apology accepted and I hope you will accept mine.
Okay, getting back to the numbers that need to be rolled, this is two components. DC and modifier. The DCs set up by the OP, and expected modifiers for characters, go hand in hand to determine what needs to be rolled. Luckily we can estimate out ranges for both of these.
For intentional tasks, it will likely be done by PCs who are good at what they do. The ranger trained in Survival and with a positive Wisdom mod tracking someone in a forest. The bard trained in persuasion. You get the idea.
That is very true, but just as often everyone in a party will be involved in a check (such as perception). Other times the best person might not be there.
Because of the DCs, it ends up that most of the roll-or-better numbers cluster more around the middle of the range. So looking at numbers that are very high is not representative of actual play.
As a matter of fact, since the variation between results is much greater than a d20 in the middle range, and much smaller at the rarely-used 17+ range you were examining, it actually flips the math and shows the opposite of common usage.
So while the math you did was good (I assume, you're good at it), the numbers you were comparing were untypical for actual play and with different distribution then the numbers used more commonly in play.
I feel the DCs were appropriate, even if the total modifiers weren't typical for PCs.
You mentioned needing a 12 or greater where both have a 45% chance. With your same 3 modifier you used (so needing a 9 or higher), d20 goes to a 60% chance vs 72% chance with 2d10. Hardly the 900% difference between then that the extreme examples found. Even the other way, needing 15 or greater is 30% vs. 21%. Both of those will be more common at the table then 17+ vs. 20+.
Here, however, you are doing the wrong comparison. I wasn't comparing the difference between 2d10 and d20 directly, I am comparing the two variants by showing the relative increase comparing someone with no modifier to someone with low modifier.
Using the 2d10 variant: For a DC 12, no modifier has a 45% of success. A +3 modifier has a 72% chance for success. That is a 60% relative increase.
Using the d20 model: For a DC 12, no modifier has a 45% of success again. A +3 modifier has a 60% chance for success. That is a 33.3% relative increase.
The point is that the 2d10 variant makes it so having a modifier will greatly increase the relative likelihood of success. You seem to be focusing on the +3 modifiers using the two systems, I'm focusing on comparing two levels of modifiers to each other and then looking at the comparison of the systems.
Perhaps your point is better illustrated with a "moderate" DC 12 and more reasonable PC-type modifiers, say +5 and + 10 (someone with proficiency and some ability versus someone with more proficiency, greater ability and/or expertise):
Using the 2d10 variant: For a DC 12, +5 modifier has a 85% of success. A +10 modifier has a 100% chance for success. That is a 17.6% relative increase.
Using the d20 model: For a DC 12, +5 modifier has only a 70% of success. A +10 modifier has a 95% chance for success. That is a 35.7% relative increase.
So, here the d20 model shows the character with the greater modifier total is more likely to succeed than the lower modifier compared to the 2d10 system. Now, that isn't to say the d20 system itself is more likely, just that the relative comparison of +5 to +10 is.
Was that more in line with your point?