The 3d6 probabilities do not, yet you've truncated them and ignored them to assume that the part you kept is similar.
Ok, since you continue to be hung up on the fact that my graph ends before the 3d6 curve gets to the top and bottom, here:
And for good measure, here's a graph of the differences in success probabilities at each adjusted DC (think of the x-axis of all of these graphs as the DC of the check minus the modifier).
So, across the range of adjusted DCs, the two methods yield success probabilities within 4.5% of each other; essentially, depending on the DC, switching from one to the other will give some characters the equivalent of somewhere between a -1 and +1.
For the reasons I keep reiterating: you've tossed data and rescaled data and recentered data.
Now hopefully we can agree that I haven't tossed any data, as I'm showing the full range of possibilities.
22 DOES NOT EXIST! There is NO mapping 22 on a d20 to ANYTHING!!!
It exists as a target, not as a possible roll. If you have a 25 AC and are facing a monster with a +3 to hit, then the adjusted DC of their roll is 22. RAW they auto-hit you on a 20, but omitting that, they need a 22 to hit you, which means
they can't.
And yet, you've compared a 0% probability for getting a 22 on d20 to a non-zero probability of getting a 16 on 3d6.
Yes. I'm comparing 0% to roughly 3% and saying those are close. You can object for gameplay reasons (essentially, using this approximation makes some tasks that would be very easy instead automatic, and some that should be very difficult impossible; or vice versa, depending on which you're treating as the reference method and which the approximation method). But it's not a mathematical error.
I am flabbergasted by this argument. You haven't limited the d20 roll to 5 and 15, you just get a 5 if you roll a 1 and a 15 if you roll a 20! All so very clear, now.
Yes, if you roll a 1, and apply 10 + (roll-10)/2 (rounding down when halving), you get 5. And if you roll a 20, and apply 10 + (roll-10)/2, you get 15. This isn't some purely theoretical exercise. You could, in principle, do that math with your rolls at the table. That's not what
@NotAYakk was actually suggesting, but adjusting the die rolls like that is mathematically equivalent to doubling your bonus and doubling the DCs' distance from 10.
when that's ignored you get your conclusion!
I haven't ignored anything. I've been entirely up front all along (as was the OP) about what happens with extreme DCs. The approximation is still good at those extremes as measured by differences in probability. You might not consider approximating 3% with 0% or vice versa to be a good approximation, and that's fine. That's a matter of gaming priorities, not math.
What you can't seem to grasp is that you've tossed at least a 1/3 of the data points to do this, meaning scaled d20 isn't similar to 3d6, it's similar to the range of 3d6 outcomes between 5 and 15 ONLY.
You keep saying this but I haven't tossed out anything.
The... actual range of possible rolls... doesn't matter when comparing actual rolls? :headdesk:
Right, because nobody is saying that 3d6 produces similar
rolls to rescaled d20 (or vice versa). We are saying that if you use a suitable rescaling that (approximately) equalizes the variance of the two distributions, then the
success probabilities are close, for
any DC you want to set.
You've swapped an unphysical model for reality and claimed victory because the math worked out.
There's nothing unphysical about any of this. It's all something you could do in your game. Either (1) roll 3d6 to resolve checks, double the result, and subtract 10. If the result ties the DC, confirm success with a d2; or (2) roll 1d20 to resolve checks, as written. The claim is that these produce very similar success probabilities, regardless of the DC.
Alternatively, if you want luck to play less of a role in your game, you can either (1) roll 3d6 to resolve checks, confirming ties with a d2; or (2) roll 1d20, halve the distance from 10 and then add 10; or (3) double all bonuses and stretch DCs to be DC' = 10 + 2*(DC - 10). (2) and (3) are exactly identical; (1) is very close, at all DCs.
Any of these are things you could actually do; they're not impractical thought experiments.