Degrees of Success with the Advantage Dice System

I forgot to ask an important question in yesterday’s post about the new dice mechanic. I spent lots of time looking at the chance of success, but never asked what the expected roll would be for the various combinations I looked at.

Well, I can’t have that, so I made a few changes to my script (you didn’t think I counted these by hand, did you?) and learned something.

PB B X H M C L
n d 2 % 3 % 4 % 5 % 6 % 7 % 8 % 9 % 10 % 11 % 12 % 13 % 14 % mean median
1 2 50.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.50 2.00
1 4 75.00 50.00 25.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.50 3.00
1 6 83.33 66.67 50.00 33.33 16.67 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3.50 4.00
1 8 87.50 75.00 62.50 50.00 37.50 25.00 12.50 0.00 0.00 0.00 0.00 0.00 0.00 4.50 5.00
1 10 90.00 80.00 70.00 60.00 50.00 40.00 30.00 20.00 10.00 0.00 0.00 0.00 0.00 5.50 6.00
1 12 91.67 83.33 75.00 66.67 58.33 50.00 41.67 33.33 25.00 16.67 8.33 0.00 0.00 6.50 7.00
1 20 95.00 90.00 85.00 80.00 75.00 70.00 65.00 60.00 55.00 50.00 45.00 40.00 35.00 10.50 11.00
2 2 100.00 25.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.25 2.00
2 4 100.00 81.25 50.00 6.25 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3.38 4.00
2 6 100.00 91.67 77.78 58.33 33.33 2.78 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4.64 5.00
2 8 100.00 95.31 87.50 76.56 62.50 45.31 25.00 1.56 0.00 0.00 0.00 0.00 0.00 5.94 6.00
2 10 100.00 97.00 92.00 85.00 76.00 65.00 52.00 37.00 20.00 1.00 0.00 0.00 0.00 7.25 8.00
2 12 100.00 97.92 94.44 89.58 83.33 75.69 66.67 56.25 44.44 31.25 16.67 0.69 0.00 8.57 9.00
2 20 100.00 99.25 98.00 96.25 94.00 91.25 88.00 84.25 80.00 75.25 70.00 64.25 58.00 13.88 15.00
3 2 100.00 100.00 25.00 12.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3.38 3.00
3 4 100.00 100.00 81.25 32.81 3.12 1.56 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4.19 4.00
3 6 100.00 100.00 94.44 80.09 54.63 15.28 0.93 0.46 0.00 0.00 0.00 0.00 0.00 5.46 6.00
3 8 100.00 100.00 97.66 91.60 80.86 64.26 40.62 8.79 0.39 0.20 0.00 0.00 0.00 6.84 7.00
3 10 100.00 100.00 98.80 95.70 90.20 81.70 69.60 53.30 32.20 5.70 0.20 0.10 0.00 8.28 9.00
3 12 100.00 100.00 99.31 97.51 94.33 89.41 82.41 72.97 60.76 45.43 26.62 3.99 0.12 9.73 10.00
3 20 100.00 100.00 99.85 99.46 98.78 97.71 96.20 94.16 91.53 88.21 84.15 79.26 73.47 15.64 16.00
4 2 100.00 100.00 100.00 62.50 12.50 12.50 6.25 0.00 0.00 0.00 0.00 0.00 0.00 4.94 5.00
4 4 100.00 100.00 100.00 74.22 24.22 10.94 1.17 0.78 0.39 0.00 0.00 0.00 0.00 5.12 5.00
4 6 100.00 100.00 100.00 94.91 77.62 39.81 8.26 3.40 0.23 0.15 0.08 0.00 0.00 6.24 6.00
4 8 100.00 100.00 100.00 98.39 92.92 80.96 59.25 24.02 3.69 1.46 0.07 0.05 0.02 7.61 8.00
4 10 100.00 100.00 100.00 99.34 97.10 92.20 83.31 68.88 47.11 15.96 1.95 0.76 0.03 9.07 9.00
4 12 100.00 100.00 100.00 99.68 98.60 96.24 91.95 84.99 74.49 59.47 38.83 11.34 1.15 10.57 11.00
4 20 100.00 100.00 100.00 99.96 99.82 99.51 98.96 98.06 96.69 94.75 92.07 88.51 83.89 16.78 18.00

As before,

  • ‘n’ indicates how many dice are being rolled;
  • ‘d’ indicates the size of the dice being rolled;
  • ‘number %’ indicates the percent chance of making that target number;
  • ‘PB’, ‘B’, ‘X’, ‘H’, ‘M’, ‘C’, ‘L’ indicate the tier the target number is associated with;
  • ‘mean’ is the average value of the dice roll;
  • ‘median’ is the median value of the dice roll (almost: when the median lands between two values I always picked the high value because I’m more interested in the high value you can expect 50% of the time rather than the low value).

I was thinking of applying a degree of success mechanism to some checks. Up to Champion tier, the median runs from exactly the target number with one die (as designed), up to four higher than the target number when rolling four dice. Groovy.

The Legendary rolls, though… the median progression goes from +0 (one die) to +7 (four dice). That’s… kind of huge, and since 4d20 maxes out at 23 you know that at least 80,000 of the 160,000 possibilities are between 18..23 (though there’s only one way to get 23, and 77 ways to get 22, and about 2,000 ways to get 21, so it’s really only 18..20 that has the rest of the… at least 78,500+ hits).

Ah well, it is supposed to be legendary, after all. If you can get 4d20 to roll I suppose you can deserve some insane degrees of success.

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