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Message: <blockquote data-quote="Esker" data-source="post: 7891579" data-attributes="member: 6966824">The issue was that 2*3d6-11 has a mean of 10, whereas 1d20 has a mean of 10.5. By introducing a "confirmation" mechanic, we're effectively reducing every roll by 1/2 (if it's easier to see, you can imagine rolling the 1d2 confirmation die after every roll and subtracting 1 on a 1, and leaving the roll alone on a 2, but this has no impact unless our roll exactly meets the DC so in practice you'd only need to roll the confirmation die in that case). This isn't reducing the likelihood to succeed by 5%; if applied to a d20 it's essentially reducing it by 2.5%, so it's as though we made the d20 have a mean of 10, like the 2*3d6-11. But that's not actually quite want we want --- I did mess that up in hastily firing off an idea that I hadn't had a chance to sit down and work through yet. What we want is to apply the confirmation die in the 3d6 case, not the d20 case. We either (1) roll 2*3d6-10, applying the confirmation mechanic, which should be comparable to an unaltered d20 system, or (2) just roll 3d6 with the confirmation mechanic, and compare that to a d20 system where bonuses are doubled together with DCs' distance from 10.Let's look at (1) first. Here are the probabilities of meeting each target natural roll from 1 to 20, comparing vanilla 1d20 (no changes to DCs or modifiers), compared to 2*3d6-10 with the confirmation correction:<img src="https://imgur.com/uxtSanm.jpg" alt="" class="fr-fic fr-dii fr-draggable " data-size="" style="" />Once we correct for the mean of 2*3d6-10 being off by a half, the approximation error is symmetric around the DC/bonus combinations with a 50% success rate. The 3d6 curve is of course nonlinear -- no points for pointing that out; everyone already recognized that -- and so the discrepancy in success rates is not constant, but it is at worst 4%.And here are the probabilities for a "reduced randomness" scheme: 3d6 with a confirmation correction and no change to DCs or modifiers, compared to 1d20 with doubled modifiers and new DCs set to 10 + 2*(DC - 10) -- that is to say, expanding around 10 by a factor of 2. (The x-axis is the target 3d6 roll; the roll needed on the d20 is different)<img src="https://imgur.com/cZYBcmU.jpg" alt="" class="fr-fic fr-dii fr-draggable " data-size="" style="" />As advertised, using 3d6 produces less variation in rolls. But if we adjust modifers and DCs we achieve the same effect with a d20. We do have a bit of an issue on the extremes, in that the scaled d20 approximation makes some rolls guaranteed or impossible, whereas they are around 96% or 4% with 3d6. But again that's something everyone has acknowledged throughout this thread, so there's nothing special in you pointing that out -- it's not a problem with anyone's math, it's a fact that distributional approximations like this have the worst fit in the tails.Note that in the latter case, the curves cross at 10.5, not at 11. This is due to the fact that by altering our DCs, tasks that previously had DC 11+modifier are no longer 50% success rate tasks; they've gotten a little more difficult, whereas the 50% success rate now sits between old DC 10+modifier (still DC 10+modifier) and DC 11+modifier (now DC 12+modifier). Introducing the confirmation mechanic on the 3d6 roll does the same thing: you no longer necessarily succeed on a DC 11+modifier task if you roll a natural 11; you only have a 50% chance.</blockquote>

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