Fantastic Friday: Probability Distribution

One of  areas which all role-playing games share (as opposed to, say, story games) is some amount of random chance affecting things. Some games have randomly-generated characteristics used to define a character. Even in games where characters aren't randomly-generated at all, dice are used in determining the results when a player attempts to overcome a challenge where the outcome (one way or the other) is not guaranteed.

For many years in the early days of the RPG hobby, most games used random numbers for character generation. Since there was a dearth of material for any one game (even the big dog in the early days, TSR, didn't originally produce scenarios -- the first scenario produced for D&D was by a company called Wee Warriors), many hobbyists converted material back and forth. This is simple to do if you're handy with math, and trivial now, when there are websites where the computer will do the work for you.

I was reading an article by Graeme Davis from Imagine #18 ("Games Without Frontiers," from September 1984), where Anthony P. Allan has worked out the probabilities for many different randomizers of the day. I love charts, so here we are. In all cases, n indicates the number of times that result comes up, p% indicates the likelihood of getting that result exactly, +P% indicates the likelihood of getting that result or higher, -P% indicates the likelihood of getting that result or lower. I found one error in going through the charts, which is fixed below.


2D6

Score n p% +P% -P%
2 1 2.778 100.000 2.778
3 2 5.556 97.222 8.333
4 3 8.333 91.667 16.667
5 4 11.111 83.333 27.778
6 5 13.889 72.222 41.667
7 6 16.667 58.333 58.333
8 5 13.889 41.667 72.222
9 4 11.111 27.778 83.333
10 3 8.333 16.667 91.667
11 2 5.556 8.333 97.222
12 1 2.778 2.778 100.000


3D6

Score n p% +P% -P%
3 1 0.463 100.000 0.463
4 3 1.389 99.537 1.852
5 6 2.778 98.148 4.630
6 10 4.630 95.370 9.259
7 15 6.944 90.741 16.204
8 21 9.722 83.796 25.926
9 25 11.574 74.074 37.500
10 27 12.500 62.500 50.000
11 27 12.500 50.000 62.500
12 25 11.574 37.500 74.074
13 21 9.722 25.926 83.796
14 15 6.944 16.204 90.741
15 10 4.630 9.259 95.370
16 6 2.778 4.630 98.148
17 3 1.389 1.852 99.537
18 1 0.463 0.463 100.000


4D6

Score n p% +P% -P%
4 1 0.077 100.000 0.077
5 4 0.309 99.923 0.386
6 10 0.772 99.614 1.157
7 20 1.543 98.843 2.701
8 35 2.701 97.299 5.401
9 56 4.321 94.599 9.722
10 80 6.173 90.278 15.895
11 104 8.025 84.105 23.920
12 125 9.645 76.080 33.565
13 140 10.802 66.435 44.367
14 146 11.265 55.633 55.633
15 140 10.802 44.367 66.435
16 125 9.645 33.565 76.080
17 104 8.025 23.920 84.105
18 80 6.173 15.895 90.278
19 56 4.321 9.722 94.599
20 35 2.701 5.401 97.299
21 20 1.543 2.701 98.843
22 10 0.772 1.157 99.614
23 4 0.309 0.386 99.923
24 1 0.077 0.077 100.000


4D6, dropping the lowest

Score n p% +P% -P%
3 1 0.077 100.000 0.077
4 4 0.309 99.923 0.386
5 10 0.772 99.614 1.157
6 21 1.620 98.843 2.778
7 38 2.932 97.222 5.710
8 62 4.784 94.290 10.494
9 91 7.022 89.506 17.515
10 122 9.414 82.485 26.929
11 148 11.420 73.071 38.349
12 167 12.886 61.651 51.235
13 172 13.272 48.765 64.506
14 160 12.346 35.494 76.852
15 131 10.108 23.148 86.960
16 94 7.253 13.040 94.213
17 54 4.167 5.787 98.380
18 21 1.620 1.620 100.000


2D10

Score n p% +P% -P%
2 1 1.000 100.000 1.000
3 2 2.000 99.000 3.000
4 3 3.000 97.000 6.000
5 4 4.000 94.000 10.000
6 5 5.000 90.000 15.000
7 6 6.000 85.000 21.000
8 7 7.000 79.000 28.000
9 8 8.000 72.000 36.000
10 9 9.000 64.000 45.000
11 10 10.000 55.000 55.000
12 9 9.000 45.000 64.000
13 8 8.000 36.000 72.000
14 7 7.000 28.000 79.000
15 6 6.000 21.000 85.000
16 5 5.000 15.000 90.000
17 4 4.000 10.000 94.000
18 3 3.000 6.000 97.000
19 2 2.000 3.000 99.000
20 1 1.000 1.000 100.000


4D5 (did anyone ever use this?)

Score n p% +P% -P%
4 1 0.160 100.000 0.160
5 4 0.640 99.840 0.800
6 10 1.600 99.200 2.400
7 20 3.200 97.600 5.600
8 35 5.600 94.400 11.200
9 52 8.320 88.800 19.520
10 68 10.880 80.480 30.400
11 80 12.800 43.200 69.600
12 85 13.600 56.800 56.800
13 80 12.800 43.200 69.600
14 68 10.880 30.400 80.480
15 52 8.320 19.520 88.800
16 35 5.600 11.200 94.400
17 20 3.200 5.600 97.600
18 10 1.600 2.400 99.200
19 4 0.640 0.800 99.840
20 1 0.160 0.160 100.000

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