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Message: <blockquote data-quote="Esker" data-source="post: 7892102" data-attributes="member: 6966824">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:<img src="https://imgur.com/mwqCNy7.jpg" alt="" class="fr-fic fr-dii fr-draggable " data-size="" style="" />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).<img src="https://imgur.com/n4ZIKPk.jpg" alt="" class="fr-fic fr-dii fr-draggable " data-size="" style="" />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.Now hopefully we can agree that I haven't tossed any data, as I'm showing the full range of possibilities.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.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.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 [USER=72555]@NotAYakk[/USER] was actually suggesting, but adjusting the die rolls like that is mathematically equivalent to doubling your bonus and doubling the DCs' distance from 10.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.You keep saying this but I haven't tossed out anything.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.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.</blockquote>

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