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:

Six scores must be at least 3;

of those, at least five must be 6 or greater;

of those, at least two must be 7 or greater;

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.