What are the Odds? (Paladins)

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I had a player who wanted to play a paladin. Rolling up a character via DMG Method IV, we struggled. When looking at the requirements for a paladin per the PHB:
Strength: 12
Int: 9
Wisdom: 13
Con: 9
Charisma: 17

According to the DMG, 1% of the population are classed, and 4.4% of available henchman are paladins. In other words, at a baseline .44% of the population are paladins. Just how hard is it to roll a legal paladin under Method I (3d6), compared to the normal population? What are the odds?

Strength: 12 (37.5%)
Int: 9 (74.1%)
Wisdom: 13 (25.9%)
Con: 9 (74.1%)
Charisma: 17 (1.9%)

Odds for rolling a legal paladin are .1%—one character in every 1,000 could be a paladin using the Method 1: 3d6!

However, at the time, we were using the requirements of the paladin as a subclass of the UA cavalier. Under Method I:
Str: 15 (9.3%)
Int: 10 (62.5%)
Wisdom: 13 (26%)
Dex: 15 (9.3%)
Con: 15 (9.3%)
Charisma: 17 (1.9%)

That _radically_ changes the chances of rolling up a paladin. The odds of rolling up a paladin with the UA requirements using Method I: 3d6? .0002%! That’s two in every million characters rolled up using Method I can be paladins.

Using the more popular Method 2: 4d6, drop lowest (also the method for rolling the aforementioned henchmen), the odds of generating a (valid) paladin are: .03%. Unlikely, and nowhere near the henchman baseline of .44%, but 1 in every 3,333 is still better than 1 in 500,000!

UA of course introduces Method V for rolling up a character, where each individual stat rolls a number of d6 depending on the relevance of that stat to the class. Nine dice are allocated to the most important stat, eight for the second-most-important, etc.  A  paladin would roll 9d6 for Charisma, and drop the lowest six rolls.

Calculating that probability involves formulas that look … insane.  Fortunately, there’s a different way of doing that—brute force.  Out of 10 million dice rolls, for 9d6, the outcomes are:

(0%) 4 -> 7
(0%) 5 -> 48
(0.01%)     6 -> 526
(.02%)     7 -> 2319
(0.07%)  8 -> 7066
(0.24%)  9 -> 23569
(0.63%)  10 -> 63392
(1.39%)   11 -> 138742
(3.01%)   12 -> 311565
(6.01%)   13 -> 600503
(10.30%) 14 -> 1029864
(15.94%) 15 -> 1594458
(22.40%) 16 -> 2240417
(22.06%) 17 -> 2205942
(17.82%) 18 -> 1781582

For the paladin the number of dice are are: 7 5 8 4 6 9 (Note: assuming no Comeliness score). Extending that brute force calculation out, the probability for the required scores for a paladin are:

Str (7d6): 15 (62.09%)
Int (5d6): 10 (92.06%)
Wisdom(8d6) : 13 (91.26%)
Dex (4d6) : 15 (23.15%)
Con (6d6) : 15 (50.92%)
Charisma (9d6) : 17 (39.88%)

Giving us a chance of rolling up a paladin of … .02%

(That doesn’t consider that under UA Method V, the odds of rolling a valid paladin are technically 100%, because if a score is below the minimum requirements, the player takes the minimum for the score).

So what does all of that tell us? Unless you meddle in one way or another with the outcome, paladins should be so vanishingly rare most GMs should see at most one paladin  in their lifetimes:

PHB Paladin, Method 1 (3d6): 1 in 1,000
PHB Paladin, Method 4 (4d6 drop lowest): 1 in every 3,333
UA Paladin, Method 5 (Arcana): 1 in 5,000
UA Paladin (Method I): 1 in 500,000 (see note above)

 

If the world population of paladins is .44% (4.4 in 1000), that also suggests that the party is more likely to encounter a paladin than have one on their own!

  • Looking at DMG Appendix C, 1% of all castles are owned by paladins. 1% of all city encounters are paladins.  2% of the 54% of all human party encounters will include 1-2 paladins.
  • According to the Monster Manual, a band of (uncommon) pilgrims are 35% likely to be Lawful good, there’s a 10% chance there will be 1-10 paladins!

 

The Mathematics Behind 4d6 Drop the Lowest

 

 

 

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