Chance of Rolling a Class

e.g. how hard is it to roll up a paladin?

It's an obvious question. What is the chance of rolling a character that qualifies for a given class?

Minimum Attributes

The chance depends on the minimum attributes for the class. Let's use the first edition values:

minimum attributes per Players Handbook (1978)

            Str Int Wis Dex Con Char
cleric:       3   3   9   3   3   3
druid:        3   3  12   3   3  15
fighter:      9   3   3   3   7   3
paladin:     12   9  13   3   9  17
ranger:      13  13  14   3  14   3
magic-user:   3   9   3   6   3   3
illusionist:  3  15   3  16   3   3
thief:        3   3   3   9   3   3
assassin:    12  11   3  12   3   3
monk:        15   3  15  15  11   3
bard:        15  12  15  15  10  15 

Odds by Class: 3d6

The original method of rolling up attributes is to use 3d6 for each attribute. The attributes are generated in the order of strength, intelligence, wisdom, dexterity, constitution, and charisma. No rearrangement is allowed.

The odds of a character qualifying for a given class by this method are:

  assassin:      7.031%
  bard:          0.002%
  cleric:       74.074%
  druid:         3.472%
  fighter:      67.215%
  illusionist:   0.429%
  magic-user:   70.645%
  monk:          0.040%
  paladin:       0.099%
  ranger:        0.176%
  thief:        74.074%

We knew that rolling up a paladin was a long shot, but rolling up a monk is harder still. Your chance of rolling up a bard are 1 in 58,140.

Odds by Class: DMG Methods

Some prestige classes might seem pointless, given how unlikely it is to roll a character that qualifies to be one. However, the DMG allows, at referee discretion, one of four methods for rolling attributes, each producing higher attributes on average. For example, in method I the player rolls the six attributes with 4d6kh3 and then rearranges the attributes as desired.

The chances of rolling up a character according to these four methods are:

  (METHOD I) 4d6kh3 PLAYER-ARRANGED:
  assassin:      93.616%
  bard:           1.581%
  cleric:       100.000%
  druid:         78.245%
  fighter:      100.000%
  illusionist:   35.782%
  magic-user:   100.000%
  monk:          13.641%
  paladin:       25.169%
  ranger:        30.470%
  thief:        100.000%

  (METHOD II)  3d6 BEST-OF-12 PLAYER-ARRANGED:
  assassin:      96.180%
  bard:           1.887%
  cleric:       100.000%
  druid:         68.206%
  fighter:      100.000%
  illusionist:   24.302%
  magic-user:   100.000%
  monk:           9.231%
  paladin:       18.704%
  ranger:        33.984%
  thief:        100.000%

  (METHOD III) 3d6 BEST-OF-6 EACH ATTRIBUTE:
  assassin:      87.053%
  bard:           3.572%
  cleric:        99.970%
  druid:         41.544%
  fighter:       99.970%
  illusionist:   10.936%
  magic-user:    99.970%
  monk:           8.487%
  paladin:        8.324%
  ranger:        29.788%
  thief:         99.970%

  (METHOD IV) BEST-OF-12-CHARACTERS 3d6:
  assassin:      58.309%
  bard:           0.021%
  cleric:       100.000%
  druid:         34.562%
  fighter:      100.000%
  illusionist:    5.024%
  magic-user:   100.000%
  monk:           0.475%
  paladin:        1.179%
  ranger:         2.097%
  thief:        100.000%

Overall, method I is the best, though method II is best for an assassin or ranger and method III is best for a bard.

Code

The code for calculating the probabilities is on GitHub.