Last Updated: 210914..230107

I have a love/hate relationship with Proficiencies. Proficiencies can be a great way to add flavor to a PC. Yet when comparing proficiencies to Class abilities, proficiencies are disproportionately effective (initial thief functions are only 25%). The very word—”proficient”—implies a high level of success.

The proficiency system in 1e biases heavily towards high stats. 1e uses a class-based system, but should all classes be equal for the same proficiencies? There’s no explicit adjustment for tasks of varying difficulty. Proficiencies don’t advance with experience. To exemplify the issue that proficiencies aren’t quite right, each edition uses a different system. Even 1e uses more than one system (Oriental Adventures *cough*).

Here’s an alternate skill system derived primarily on the Dragon Magazine #225 system, and vaguely similar to OA with different checks. Skills are weaker to start, but improve as a PC levels.

A character starts with a certain number of skills, which can be improved or expanded upon as the character levels. When performing a skill check, the player rolls a d20, and is successful if they roll equal to or less than the target number. Skill checks are modified for the difficulty of the task, but difficulty of the task is determined based on the PC’s skill level.

• Standard proficiency slots per class + Intelligence bonus determine Skill Slots available to acquire or improve new skills.
• A specific skill provides a base 10 target number +/- a skill modifier, representing the intrinsic difficulty or simplicity of that particular skill (e.g., Alchemy, based on Intelligence, has a -3 target number penalty, i.e., a base target number of 7).
• The ability score associated with the particular skill modifies the target number:

Ability score modifier
12	+1
13	+1
14	+1
15	+2
16	+2
17	+3
18	+3

• Non-Fighters have a -1 penalty on STRENGTH proficiencies, and a +2 max bonus on CONSTITUTION skills (aligning Strength checks more closely to the ability tables).

• Learning a skill that mimics a class skill from another class incurs an additional -6 target number penalty.
• Multi-class characters start with a number of skills equal to the total proficiency slots from both classes, and earn new skill slots at the proficiency rate of the fastest class.
  • Specializing in a skill (using an additional slot) improves the base target number by 3 (journeyman).
  • Double specializing (spending two additional slots) will improve the skill target number by an additional 2 (master). Further specialization will improve the skill target by 1 for each additional point spent.
• Each level the PC gains 1 additional point to improve/increase one previously-selected skill’s target number of their choice by 1 point (multi-class characters get 1 each time they reach a new highest level).
  • The player must choose which skill will be modified upon gaining the new level.
  • Unlike gaining new slots, the experience modifier may not be saved from level to level. If they are not assigned, they are lost.
  • The player may select any of the character’s previously-selected skills to improve. The only restriction on this choice is that a newly-acquired skill cannot be chosen.
  • The skill target number is then modified further by the associated ability score.
• Attempts can be made against an unlearned skill, with base target number of 0.
• When two characters with the same skill work together on the same task, use the score of the player with the best skill (the one with the greatest chance of success). Furthermore, a minimum of +1 bonus is added for the other character’s assistance (+1 for each additional point they have added to the skill). The bonus can never be from more than 1 PC; too many assistants are sometimes worse than having none.
• To determine success, Roll d20 <= the modified target number based on the difficulty of the particular task for someone who knows the skill:

Modifier to Target number
Easy	+5
Average	0
Hard	-5
Very hard	-10
Nearly impossible	-15

• Always fail on a 20. Success on a 1 is not guaranteed.
• When there are opposed skill checks, use the delta of success to determine success.
• Where applicable, acquiring a skill conveys the initial expectation of apprentice-level ability, not mastery. In other words, determine the difficulty as if the PC was an apprentice, not a master. If the PC has spent 2 proficiency slots, then determine difficulty as if the PC were a journeyman, and 3 (or more) slots as if they were a master.

Optional*

Buying a non-class skill costs 1 more slot. Class skills are a combination of DRMG 225 and 2e, leaning towards permissive.

F	Fighter	Paladins are also Clerics, Rangers are also Magic Users, Barbarians are also Thieves
M	Magic User
C	Cleric	Druids are also Fighters
T	Thief
B	Bard
K	Monk	Monks are also Thieves

^*I apply this only to skills that replicate class-like abilities.

Bibliography

TSR 2018, Oriental Adventures [1e], 1985.
TSR 2019, Dungeoneer’s Survival Guide [1e], 1986.
TSR 2020, Wilderness Survival Guide [1e], 1986.
Dragon Magazine, Issue 224, “Back in the Saddle (Again) January 1996.

 

At 10th Level (Master Thief), thieves are able to decipher magical writings and utilize scrolls of all sorts, excluding those of clerical, but not druidic, nature. However, the fact that thieves do not fully comprehend magic means that there is a 25% chance that writings will be misunderstood. Furthermore, magic spells from scrolls can be mispronounced when uttered, so that there is an increasing chance per level of the spell that it will be the reverse of its intent (PHB, p. 27)

I’ve got two thieves approaching level 10—it’s time to refresh myself on thieves casting from scrolls. To translate the above, 10th+ level thieves can read magic user, illusionist, and druidical spells, in addition to the Protection scrolls usable by any class (DMG, p. 128: “Protection scrolls can be read by any class or race of character even without a magic spell”).  While Intelligence can limit spell casting, “Those characters able to read and employ scroll spells may do so regardless of other restrictions” (DMG, p. 128), so the Intelligence of the thief doesn’t limit the level of spells .

The level that rangers and paladins cast a level 1 spell can be a bit confusing (“Does my Level 9 paladin cast Level 1 cleric spells as Level 1, or Level 9?”) but there’s no such question for scrolls. Unless specified otherwise, the scroll will function as a caster 1 level higher than required to cast the spell for the associated class, but at least 6th level (DMG, p. 128). A fireball (Level 3) can be cast as a level 5 magic-user, so will cast as if by a level 6 magic-user (which is also the minimum, i.e., Level 1 and Level 2 magic-user spells will also cast as if by a level 6 magic-user).

So the only thing left is “There is a 25% chance that writings will be misunderstood, and [an increasing chance of a reverse effect].”  We have only one source for chances of magic spell failure:

DMG, pg. 128: Magic Spell Failure: If a spell-user acquires a scroll with a spell(s) of a level(s) not yet usable by the character, the spell-user may still attempt to use the spell; the chance of failure, or other bad effect, is 5% per level difference between the character’s present level and the level of magic use at which the spell could be used. [Any score that represents failure then uses the following table]

Level Difference	Total Failure	Reverse or Harmful Effect
1-3	95%	5%
4-6	85%	15%
7-9	75%	25%
10-12	65%	35%
13-15	50%	50%
16 and up(a)	30%	70%

(a) Aside: Why isn’t just 16-17? Even assuming 0-level casters, 16-18?

For other spell-users, the chance of magic spell failure is “5% per level difference between the character’s present level and the level of magic use at which the spell could be used”. For the thief, it’s a flat 25% (“there is a 25% chance that writings will be misunderstood “). That makes a strange sort of sense, since the thief doesn’t have a difference in “level of magic use at which the spell could be used” since it’s always Level 10. The implication of Magic Spell Failure is that failure is determined at the time of reading the scroll—the thief won’t know if they will be successful until they attempt to use the scroll. And once they read the scroll to release its magic (successful or not), the writing completely and permanently disappears from the scroll (DMG, p. 128).

But that still leaves the question of how to determine the “increasing chance per level of the spell that it will be the reverse of its intent.” The Magic Spell Failure Table uses the Level Difference to determine the chance of a reverse effect. However, all levels of spell are theoretically accessible by the thief; at 10th level they can utilize “scrolls of all sorts”. Are we back to the paladin/ranger conundrum? However, the thief’s chance of reversal is tied to spell level, not character level. We also don’t consider the thief level for chance of failure—why would we use it for chance of reversal?

Now with no By-the-Book reference, how then to determine the  chance of reversal? The most extreme chance of reversal on the Magic Spell Failure Table is 70%. We can expect that even level 1 spells have a chance of reverse effect for the thief. Using the Magic Spell Failure Table for inspiration, we can come up with several methods to determine the chance of a reverse effect upon failure:

Spell Level	Method: Level Difference	Actual Reverse Frequency		Method: By 5%	Actual Reverse Frequency		Method: By 10%	Actual Reverse Frequency
1	5%	1.25%		5%	1%		10%	3%
2	5%	1.25%		10%	3%		20%	5%
3	5%	1.25%		15%	4%		30%	8%
4	10%	2.50%		20%	5%		40%	10%
5	10%	2.50%		25%	6%		50%	13%
6	10%	2.50%		30%	8%		60%	15%
7	15%	3.75%		35%	9%		70%	18%
8	15%	3.75%		40%	10%		80%	20%
9	15%	3.75%		45%	11%		90%	23%

Actual Reverse Frequency (above) is the calculated likelihood of a reverse effect occurring each time a thief uses a scroll (e.g., 25% chance of failure × 10% chance of reversal = 2.5% chance of the reverse effect occurring). From there we have to consider what was intended. Should a thief have, worst case, a 3.75% chance of a reverse effect when using a 9th level scroll? Probably not. 11% still seems too low. But a 23% total chance of a 9th level spell occurring in reverse seems reasonable, and comes close to maxing out the base chance of spell reversal at 90% (and appropriately worse than the 70% for reverse effect for the Magic Spell Failure Table). That leaves us with the following table:

Thief Spell Failure
Spell Level	Chance of Failure	Chance of Reversal on Failure
1	25%	10%
2	25%	20%
3	25%	30%
4	25%	40%
5	25%	50%
6	25%	60%
7	25%	70%
8	25%	80%
9	25%	90%

Now that that’s settled (inasmuch as it can be), should the thief be able to cast spells from any spell book? “In extremis, the DM may allow a magic-user to cast a spell directly from any sort of spell book just as if the book were a scroll” (UA, p. 80). The thief can only explicitly “utilize scrolls”, and the ability to cast a spell directly from a spell book is an ability granted to the magic-user “as if it were a scroll”—not that spell books are scrolls. So no.

For assassins, tertiary functions of assassins are the same as thieves, performing thieving at two levels below their assassin level (PHB., p. 29; UA, p. 13). So 12th Level assassin functions the same as a 10th level thief. And, like the thief, their ability to read scrolls never improves.

 

Last Updated: 220204 … 220611 (heavily revised)

Back stabbing is the striking of a blow from behind, be it with club, dagger, or sword. The damage done per hit is twice normal for the weapon used per four experience levels of the thief, i.e. double damage at levels 1-4, triple at 5-8, quadruple at levels 9-12, and quintuple at levels 13-16. Note that striking by surprise from behind also increases the hit probability by 20% (+4 on the thief’s “to hit” die roll).

There’s a lot of uncertainty and misunderstanding about the relative “power” of back stab.  As the thief’s abilities often boil down to back stab, many DMs limit thief abilities. So just how good is back stab, really?

Let’s perform a reasonably direct comparison to the fighter:

• A fighter, 0 XP, Level 1, Strength 17, THAC0 20 –1 (Str), armed with a long sword (average 4.5 points of damage +1 (Str))
• A thief, 0 XP, Level 1, Dexterity 17, THAC0 20–4 (back stab), armed with a longsword (average 4.5 points of damage × 2 (back stab)) ). MS: 15%+5% (Dex), HiS: 20% + 5% (Dex).
• Our generic monster, AC: 5.

Both have a 17 in their Prime ability (Dexterity and Strength respectively), increasing the fighter’s To Hit and Damage, and the thief’s chance to MS and HiS. The thief’s backstab, on average, decreases an opponent’s AC by 1 (no Dex or shield bonus).

In a discussion of THAC0, you have to consider the early repeating 20 in the thief’s to hit chart, and adjust appropriately. A further consideration when comparing classes is relative level for commensurate XP. Generally, a thief should be one level higher than a fighter starting at level 3:

THAC0

Level	1	2	3	4	5	6	7	8	9	10	11	12
Fighter	19	19	17	17	15	15	13	13	11	11	9	9
Thief (+4)*	15	15	15	15	13	13	13	13	10	10	10	10
T (+4,lvl adjusted)*	15	15	15	13	13	13	13	11	11	11	11	9
* Adjusted for the thief’s 20,20,19 … in the THAC0 chart, Str bonus, Dex benefit lost.

At low levels, the +4 bonus to hit makes a thief more likely to hit than the fighter, but the fighter’s more rapid THAC0 advance overwhelms the +4 bonus by level 7.

Number of Attacks

While the fighter attacks (at least) every round, back stab is limited in frequency. For the thief to back stab, they must approach, back stab, then retreat in order for their opponent to be no longer aware of their presence. Assuming that takes but 2 rounds:

Attacks / Level	1	2	3	4	5	6	7	8	9	10	11	12
Fighter	1	1	1	1	1	1	3/2	3/2	3/2	3/2	3/2	3/2
Thief	.5	.5	.5	.5	.5	.5	.5	.5	.5	.5	.5	.5

Damage

The thief gains a damage multiplier on back stab, but fewer attacks/round. If we assume both the fighter and thief are using a long sword (average 4.5 hp of damage), then for each hit our fighter does 5.5 hp damage, where the thief using back stab will average 9 damage (levels 1-4), 13.5 damage (levels 5-8), or 18 (levels 9-12).

Putting the Pieces Together

Now, considering our “average” AC:5 monster, that brings us to a simplistic chance to hit per attack of:

Chance to hit AC: 5	1	2	3	4	5	6	7	8	9	10	11	12
Fighter (Str +1)	35%	35%	45%*	45%*	55%*	55%*	65%*	65%*	75%*	75%*	85%*	85%*
T (lvl adj. +4 back)	70%	70%	70%	80%	80%	80%	80%	90%	90%	90%	90%	100%

* Ignoring that our fighter will almost certainly specialize. And again…

Given our fighter also has +1 to damage from strength, the average damage per attack is:

Damage/attack	1	2	3	4	5	6	7	8	9	10	11	12
Fighter (Str +1)	1.9	1.9	2.5	2.5	3.0	3.0	3.6	3.6	4.1	4.1	4.7	4.7
Thief (lvl. adj w/ mult)	6.3	6.3	6.3	7.2	10.8	10.8	10.8	12.2	16.2	16.2	16.2	18.0
Damage ratio/attack	3.3	3.3	2.5	2.9	3.6	3.6	3.0	3.4	3.9	3.9	3.5	3.9

Or, put another way, the thief’s back stab averages 3.4 times the damage of a fighter hit-for-hit. Not bad! However, while the thief can only (best case) back stab once every two rounds, the fighter’s attacks per round increase, and the thief’s relative chance to hit decreases. A more accurate ratio looks more like:

Damage/round	1	2	3	4	5	6	7	8	9	10	11	12
Fighter (Str +1)	1.9	1.9	2.5	2.5	3.0	3.0	3.6	3.6	4.1	4.1	4.7	4.7
T (lvl. adj)	3.2	3.2	3.2	3.6	5.4	5.4	5.4	6.1	8.1	8.1	8.1	9.0
Damage/round	1.6	1.6	1.3	1.5	1.8	1.8	1.5	1.7	2.0	2.0	1.7	1.9

Using back stab most generously (every other round), the thief averages 1.7 times damage as a fighter. Nothing earth-shattering—if the thief can back stab every other round, he’s in the ball park of a fighter, but far more vulnerable (AC, HP) once engaged in melee. Hopefully their opponent lets them withdraw. Otherwise the thief remains locked in combat after the initial back stab attempt, with his worse hit points, THAC0 and armor class—not ideal.  Or flee, providing their opponent a +4 bonus in return!

The GM likely requires the thief to move around to their opponent’s back again, for an additional round for the opponent to “lose awareness” (assuming their opponent doesn’t pursue!), bringing that ratio down to 1.1 times the fighter.

Skill Checks

The Level 1 fighter requires a 14 To Hit, so: 35% (TH) * 5.5 Damage (Str) = 1.9 DPR.

Assume the best case: back stab every other round, and no other checks:

Our Level 1 thief requires an 10 To Hit, and: (45% (TH) * 9 Damage (Back stab) ) = 2 DPR.

In other words, back stab DPR looks awesome—better than the fighter! Back stab includes one more consideration—thief abilities. Move Silently is used “to approach to back stab”. Failure means that movement was not silent, and thus back stab fails. The ability to move silently ranges from 10% to 99%, but even a 9th level thief will only succeed 70% (barring race and Dexterity modifiers). That brings our thief back stab / two round damage number down to 50% of the fighter:

Damage/round w/ MS	1	2	3	4	5	6	7	8	9	10	11	12
Fighter (Str +1)	1.9	1.9	2.5	2.5	3.0	3.0	3.6	3.6	4.1	4.1	4.7	4.7
Thief (lvl. adj)	1.1	1.6	1.6	1.8	2.7	2.7	2.7	3.0	4.1	4.1	4.1	4.5
Move Silently (level adj)	0.2	0.3	0.3	0.4	0.5	0.6	0.6	0.7	0.8	0.9	0.9	1.0
Damage ratio/round	0.1	0.2	0.2	0.3	0.4	0.5	0.5	0.6	0.8	0.8	0.8	1.0

(For other comparisons, the chance to Hide in Shadows is worse than Move Silently)

In other words, a lot of math later, on a round-by-round basis, the thief, using his premium back stab ability, with back stab every other round, averages 50% of the damage of a fighter. To back stab, he further exposes his poorer AC and hit points to attack. Under the assumption he’s using a long sword, instead of the perhaps more frequent club or dagger.

Multiple Skill Checks:

Some GMs require MS and HiS rolls instead of, or in addition to Surprise. If the GM requires MS (or HiS depending on the GM), we get to the meat of the issue:

Fighter melee DPR (d8)	1.9
Thief back stab DPR	2.03
Thief back stab DPR w/ MS	0.41
Thief back stab DPR w/ HiS	0.30
Thief back stab DPR w/ MS × 2	0.08
Thief back stab DPR w/ His + MS	0.06

With only a single MS check, the fighter averages 429% more damage. Should the GM require two successive checks, the thief’s back stab DPR drops through the floor—the fighter averages 20× more damage than a thief with back stab! If you ask yourself, “Why do my thieves never back stab?”, this would be why.

Surprise

By the Book, Surprise—not MS, not HiS—is required for back stab. Let’s evaluate the thief’s back stab DPR with Surprise. The default chance for Surprise is 2 in 6. A silent thief increases that range to 3 in 6. A silent hidden thief increases that Surprise range to 4 in 6.

“Assume the party of characters, moving silently and invisibly, comes upon a monster. They have 4 of 6 chances to surprise.” PHB, p. 103.

With a generous GM, a thief approaching from behind could be considered invisible (at least hard to see since the thief is by definition at their opponent’s back), receiving the benefit of being hidden after a successful MS.

The Level 1 thief’s back stab DPR under Surprise:

Thief Back stab DPR w/ Surprise (2 in 6) + To Hit	0.60a
Thief Back stab DPR w/ Surprise (w/ MS bonus) (3 in 6) + To Hit	0.76a
Thief Back stab DPR w/ Surprise (w/ MS bonus + auto Hide from rear) (4 in 6) + To Hit	1.07a

^aSurprise includes a 16% chance for one attack, a 16% chance for two additional attacks, and (potentially) chances for 3 or even 4 attacks depending on the modified Surprise roll.  Additional back stabs don’t multiply damage, and are only +2 to hit—see Polyhedron #31, p. 29.

If the L1 thief rolls Surprise for a successful back stab, their DPR is .6. If you incorporate a MS check before the Surprise roll to increases the chance of Surprise to 3 in 6, the DPR increases. If you assume that MS includes being hidden (hard to see someone approaching successfully from behind), then their DPR is better yet. Our thief’s DPR is less than that of the fighter, but using Surprise is better than a single MS check.

And the same comparison with our PCs leveled up to the thief’s triple damage? One can quibble about the thief’s speed of advancement for levels, but the levels for the two classes still overlap in this range, so any difference between the two classes is brief.

A thief, 19,000 xp, Level 5, Dexterity 17, THAC0 19 – 4 (back stab). MS: 40%+5% (Dex), HiS: 31% + 5%(Dex), armed with a longsword (average 4.5 points of damage × 3 (Back stab))

A fighter, 19,000 XP,  Level  5, Strength 17, THAC0 16 –1 (Str), armed with a long sword (average 4.5 points of damage +1 (Str))

Here’s all the data in totality:

Prime: 17, Long sword, Opponent AC:5	Level 1	Level 5
Fighter Melee DPR	1.93	3.03
Thief Melee DPR	1.13	1.58
Thief Back stab DPR (only To Hit, no other mods)	2.03	3.71
Thief Back stab DPR w/ MS + To Hit	0.41	1.67
Thief Back stab DPR w/ HiS + To Hit	0.30	1.34
Thief Back stab DPR w/ MS + MS + To Hit	0.08	0.75
Thief Back stab DPR w/ His + MS + To Hit	0.06	0.60
Thief Back stab DPR w/ Surprise (2 in 6) + To Hit	0.60	0.96
Thief Back stab DPR w/ Surprise (w/ MS bonus) (3 in 6)  + To Hit	0.76	1.41
Thief Back stab DPR w/ Surprise (w/ MS bonus+auto Hide from rear) (4 in 6) + To Hit	1.07	2.32

 

Other considerations should you desire more math:

• A 1e fighter likely has additional bonuses to hit beyond a base 17 Strength. If the campaign uses Specialization, the fighter will have additional bonuses to hit in addition to additional attacks / round.
• At L6 the fighter’s attacks go to 3/2 without specialization (or 2/1 with specialization), increasing DPR by 50%.
• If the GM limits the thief to only one back stab every three rounds, all of the thief DPRs decrease 33%.
• If the attack renders a creature dead, who cares if you compare a back stab 10 hp / round vs 4 hp / round when the creature has only 4 hp?
• A good argument can be made that Surprise can, at most, provide only 2 additional attacks, as defined as “Surprise” (1 segment) and “Complete Surprise” (2 segments), where the above assumes the possibility of up to 4 segments of Surprise.

The net result is that calculating back stab damage by Surprise has a higher DPR for the thief, yet still considerably less than a fighter of comparable level. Calculating back stab taking more than two rounds, or requiring multiple rolls for success, almost guarantees futility for the thief and back stab.

 

And of course, back stab isn’t always an option. Certain creatures (otyughs, slimes, molds, etc.) either negate surprise or have no definable “back”, thus negating this ability. Back stab isn’t possible against those “aware” of the thief. Depending on the GM, the opponent could be deemed aware after the first attack, so only once per melee. Results swing further when you start to include a more usual fighter to hit/damage bonus from Strength.

UA further lessened back stab with weapon specialization providing fighters a 50% increase in number of attacks (and thus a 50% increase in damage), an increased chance to hit, and a ≥ 20% increase in damage (with at least +1 damage from weapon specialization). And that’s ignoring specializing more than once.

Recommendations

Don’t, just don’t, require multiple skill checks.  Use Surprise, improved by MS (or HiS).

Don’t fear the thief getting to back stab more than once in a fight. The time he spends moving around limits the overall damage per round.

While +4 to hit looks good, over levels it loses value in comparison to the fighter’s THAC0 table. The thief’s bonus to hit should increase at the same rate as the damage multiplier. While you could permit a thief to specialize, that’s not quite the advantage it appears given the thief’s fewer proficiency slots. A flat additional +1 to hit at level 5-8, +2 to hit at levels 9-12 makes back stab scale better, especially if playing with weapon specialization.

While back stab has the occasional wild swing (rolling max damage with a 4× multiplier), the reverse is also true when the high-level thief rolls a “1” for 4 points of damage. Back stab is hard to pull off, and exposes the thief to significantly more peril. If the back stab is successful, a minimum of damage would help offset the risk—a 2-point minimum at level 5, a 3-point minimum at level 9, etc.

210501

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