Probability Of An Adventurer's Ability Scores

[Great Ax]

How likely is a being possessing the standard array (it may be different, but I use 15, 14, 13, 12, 10, 8) within the D&D universe? Creating a bell curve for the probability of the possible results of 3d6, I got this:

3: 0.46%
4: 1.4%
5: 2.8%
6: 4.6%
7: 6.9%
8: 9.7%
9: 11.6%
10: 12.5%
11: 12.5%
12: 11.6%
13: 9.7%
14: 6.9%
15: 4.6%
16: 2.8%
17: 1.4%
18: 0.46%

CORRECT ME IF I AM WRONG- Using this equation (A% x B% = C%), I got this: .046 * .069 * .097 * .116 * .125 * .097 = .0000004 (.00004%). This number is for the human population, as ability score bonuses and different population sizes would effect the probability of scores. This means that if there are as many humans in the D&D universe as there are in our own (~7.5 billion), there are only 3,000 individuals with this set of scores. When rolling scores, obviously you could get scores as good as not having a score below 14 for example, 15 even (don't even go as far as all 18s... the numbers get scary). This would pretty much make you a god among men as far as raw ability goes, being one of a literal handful of people in the vast human population (or maybe the only one).

Now think about the adventuring party as a whole. What are the odds that 4 - 5 people possessing these unbelievable abilities meet up and spend the next several years with each other killing things and taking money from incompetent people and corpses...

[eggynack]

Not really in the mood to do a bunch of probability, but at a baseline, you're missing two important factors. First, order doesn't matter. Your current evaluation is assuming the numbers have to be in that exact order, but they can be in any order. Second, you're assuming that people need these exact scores, and they don't. Even if you need these numbers to do well, it's obvious that you can have at least these numbers and do well. In other words, the probability associated with each number should be the sum of that percentage with every percentage above it. And, of course, you don't need exactly these numbers to do well. You can have -1 for one stat and +1 for another, or subtract one from every odd value, or just generally consider things in the form of point buy, and you'll do quite well.

[OldTrees1]

Since the array 15/14/13/12/10/08 contains no duplicates there are 6! (720) ways to arrange it. So the probability of a random human being one with that array is:

6! * .046 * .069 * .097 * .116 * .125 * .097 = 0.03% for that 1 array.

However perhaps also interesting what is the probability a random human only has ability scores between 08-15:
0.791^6 = 24.49%. So a quarter of the population falls in that 08-15 middle.

Also possibly interesting, the probability that a random human has 1 08, 1 15, and 4 stats between 08-15 is:
(6!/4!) * 0.097 * 0.046 * 0.791^4 = 5.24%

[Coventry]

"Order doesn't matter" means you multiply that percentage by (6*5*4*3*2*1) ... 720. There are six places that 15 might have come from, each having 5 places that 14 could have come from, et cetera.

Still a low percentage, but the "average" person doesn't try to be an adventurer, any more than an average person goes to the Olympics.

[Coidzor]

How likely is a being possessing the standard array (it may be different, but I use 15, 14, 13, 12, 10, 8) within the D&D universe? Creating a bell curve for the probability of the possible results of 3d6, I got this:

Monsters are assumed to have completely average (or standard) ability scores—a 10 or an 11 in each ability, as modified by their racial bonuses.

[...]

The elite array is: 15, 14, 13, 12, 10, 8.

[...]

The nonelite array is: 13, 12, 11, 10, 9, 8.

So we're asking about the probability of possessing the Elite Array. Which is appropriate for basically any NPC with levels in PC classes.

IIRC, the Standard Array is actually 11, 11, 11, 10, 10, 10 for 3 odd, 3 even ability scores unless a creature or template has an odd ability score adjustment to it.

[WhatThePhysics]

Assuming equal odds of any two people meeting, there's a 1-in-28 septillion chance 4 elite array characters will meet, and 1-in-65 nonillion chance 5 will meet.

Assuming elite array characters are a million times more likely to meet each other, there's a 1-in-28 chance 4 will meet, and 1-in-65 chance 5 will meet.

[Remuko]

If were to assume these people using this array are PCs then in 3.0 and 3.5 the method for determining stats is 4d6 drop the lowest which should alter the results for any given number (upping the average without changing the min or max results).