Qualifying for character classes in Advanced Dungeons & Dragons

When rolling ability scores in AD&D, particular minimum values must be achieved in order to qualify for each race and class. We will show by an example how one may calculate the probability of meeting these requirements; generalizing the method will be left for the interested reader.

Consider the Human Fighter, which needs the following minimum scores: Strength 9, Intelligence 3, Wisdom 6, Dexterity 6, Constitution 7, Charisma 6.

Let $P(a)$ denote the probability of rolling a single score of exactly $a$ and let $P(a,b)$ denote the probability of rolling a single score of at least $a$ and at most $b$.

If the scores must be rolled in order, then the probability of qualifying for the class is found by multiplying together the probabilities of meeting or exceeding each minimum value: $$P(3,18)P(6,18)^3P(7,18)P(9,18).$$

For 3d6 that works out to about 58.3 percent.

If the rolling order doesn't matter, then the problem is more complex. Observe that we have the following requirements:

1. Six scores must be at least 3;
2. of those, at least five must be 6 or greater;
3. of those, at least two must be 7 or greater;
4. and of those, at least one must be nine or greater.

Let $i$ be the number of scores of nine or higher. If $j$ is the maximum number of scores which are at least 7 but less than 9, then for any $i$ the number of such scores will be $j-i$. Similarly, if $k$ is the maximum number of scores which are at least 6 but less than 7, then for any $j$ the number of such scores will be $k-j$. That leaves $6-k$ scores which are at least 3 but less than 6.

The probability of rolling $i$ scores of probability $P(9,18)$, $j-i$ scores of probability $P(7,8)$, $k-j$ scores of probability $P(6)$, and $6-k$ scores of probability $P(3,5)$ has a multinomial distribution: $$\frac{6!}{i!(j-i)!(k-j)!(6-k)!}P(9,18)^iP(7,8)^{j-i}P(6)^{k-j}P(3,5)^{6-k}.\qquad\textrm{(1)}$$

Now from the requirements above, $k$ can vary from 5 to 6, so it follows that $j$ may vary from 2 to $k$, and hence $i$ can vary from 1 to $j$. Summing $\textrm{(1)}$ for all these variations, we find the total probability of qualifying for the Fighter class: $$\sum_{k=5}^6\sum_{j=2}^k\sum_{i=1}^j\frac{6!}{i!(j-i)!(k-j)!(6-k)!}P(9,18)^iP(7,8)^{j-i}P(6)^{k-j}P(3,5)^{6-k}.$$

For 3d6 that works out to about 97.14 percent.

    
Minimum scores
    
Unordered
    
Ordered
Character
 
Str
Int
Wis
Dex
Con
Cha
 
3d6
 
3/4d6
 
3d6
 
3/4d6
Cleric
6
6
9
3
6
6
97.140%
99.805%
61.280%
85.434%
Druid
6
6
12
6
15
6
30.868%
73.455%
2.873%
13.622%
Fighter
9
3
6
6
7
6
97.140%
99.805%
58.305%
84.033%
Paladin
12
9
13
6
9
17
3.966%
24.171%
0.094%
1.378%
Ranger
13
13
14
6
14
6
3.256%
29.351%
0.161%
2.927%
Magic-user
3
9
6
6
6
6
97.140%
99.805%
61.280%
85.434%
Illusionist
6
15
6
16
3
6
7.562%
35.744%
0.372%
2.915%
Thief
6
6
3
9
6
6
97.140%
99.805%
61.280%
85.434%
Assassin
12
11
6
12
6
3
58.571%
93.483%
6.395%
27.134%
Monk
15
6
15
6
11
6
7.625%
39.576%
0.372%
3.781%
Bard
15
12
15
15
10
15
0.027%
1.581%
0.002%
0.146%
Half-elf Cleric
6
6
9
6
6
6
75.236%
93.253%
58.443%
84.445%
Half-orc Cleric
5
6
9
3
12
8
92.909%
99.643%
21.788%
51.230%
Half-elf Druid
6
6
12
6
15
6
30.868%
73.455%
2.873%
13.622%
Dwarf Fighter
9
3
6
6
11
7
95.464%
99.763%
30.568%
62.123%
Elf Fighter
9
8
6
6
8
8
73.219%
93.089%
39.643%
73.306%
Gnome Fighter
9
7
6
6
8
6
75.222%
93.253%
48.858%
79.235%
Half-elf Fighter
9
4
6
6
7
6
94.949%
99.368%
58.035%
83.968%
Halfling Fighter
10
6
6
7
10
6
73.678%
93.193%
30.747%
63.877%
Half-orc Fighter
8
3
6
6
12
8
91.247%
99.481%
23.950%
53.550%
Half-elf Ranger
13
13
14
6
14
6
3.256%
29.351%
0.161%
2.927%
Elf Magic-user
3
9
6
6
7
8
97.089%
99.804%
51.230%
80.163%
Half-elf Magic-user
3
9
6
6
6
6
97.140%
99.805%
61.280%
85.434%
Gnome Illusionist
6
15
6
16
8
6
6.269%
33.908%
0.312%
2.749%
Dwarf Thief
8
6
3
9
11
7
94.713%
99.731%
26.858%
59.262%
Elf Thief
6
8
3
8
7
8
96.171%
99.771%
50.920%
80.558%
Gnome Thief
6
7
3
9
8
6
97.089%
99.804%
51.230%
80.163%
Half-elf Thief
6
6
3
9
6
6
97.140%
99.805%
61.280%
85.434%
Halfling Thief
7
6
3
8
10
6
96.883%
99.802%
43.225%
73.874%
Half-orc Thief
5
6
3
9
12
8
92.909%
99.643%
21.788%
51.230%
Dwarf Assassin
12
11
6
12
11
4
32.381%
79.058%
3.337%
20.044%
Elf Assassin
12
11
6
11
7
8
51.151%
88.806%
6.798%
29.827%
Gnome Assassin
12
11
6
12
8
3
57.852%
93.361%
5.619%
25.884%
Half-elf Assassin
12
11
6
12
6
3
58.571%
93.483%
6.395%
27.134%
Half-orc Assassin
11
11
6
12
12
5
31.003%
78.002%
3.291%
19.982%
Half-elf Bard
15
12
15
15
10
15
0.027%
1.581%
0.002%
0.146%
Half-elf Cleric/Fighter
9
6
9
6
7
6
75.038%
93.249%
43.189%
75.215%
Half-orc Cleric/Fighter
8
6
9
6
12
8
69.795%
92.831%
17.741%
47.931%
Half-elf Cleric/Fighter/Magic-user
9
9
9
6
7
6
73.268%
93.152%
33.545%
68.110%
Half-elf Cleric/Ranger
13
13
14
6
14
6
3.256%
29.351%
0.161%
2.927%
Half-elf Cleric/Magic-user
6
9
9
6
6
6
75.042%
93.250%
45.393%
76.468%
Half-orc Cleric/Thief
5
6
9
9
12
8
79.540%
97.012%
16.139%
45.854%
Half-orc Cleric/Assassin
11
11
9
12
12
8
17.116%
59.696%
2.182%
17.127%
Elf Fighter/Magic-user
9
9
6
6
8
8
73.137%
93.087%
35.044%
69.587%
Half-elf Fighter/Magic-user
9
9
6
6
7
6
75.038%
93.249%
43.189%
75.215%
Gnome Fighter/Illusionist
9
15
6
16
8
6
6.210%
33.885%
0.242%
2.489%
Dwarf Fighter/Thief
9
6
6
9
11
7
72.650%
93.127%
22.643%
55.604%
Elf Fighter/Thief
9
8
6
8
8
8
63.313%
90.632%
34.832%
69.930%
Gnome Fighter/Thief
9
7
6
9
8
6
74.781%
93.237%
37.948%
71.751%
Half-elf Fighter/Thief
9
6
6
9
7
6
75.038%
93.249%
43.189%
75.215%
Halfling Fighter/Thief
10
6
6
8
10
6
73.575%
93.189%
28.394%
61.950%
Half-orc Fighter/Thief
8
6
6
9
12
8
69.795%
92.831%
17.741%
47.931%
Elf Fighter/Magic-user/Thief
9
9
6
8
8
8
63.286%
90.631%
30.791%
66.382%
Half-elf Fighter/Magic-user/Thief
9
9
6
9
7
6
73.268%
93.152%
33.545%
68.110%
Elf Magic-user/Thief
6
9
6
8
7
8
74.924%
93.240%
42.928%
75.586%
Half-elf Magic-user/Thief
6
9
6
9
6
6
75.042%
93.250%
45.393%
76.468%
Gnome Illusionist/Thief
6
15
6
16
8
6
6.269%
33.908%
0.312%
2.749%

See part two to find probabilities for multiple attempts.