D&D Statistics

I'm the kind of person who likes to know the odds. When I started playing in a D&D (3rd edition) game I looked online to see if anyone had broken down the various rolls. Unfortunately I couldn't find anything, so I decided to write my own.

Be warned! I am competent with computers and basic mathematics, but I'm no statistics expert. You can trust these numbers but not necessarily any conclusions I may draw from them.

Kevin Sullivan <kevin@klubkev.org>

Chance of winning a contest of skills
Diff Win Lose Tie
+0 47.5% 47.5% 5.0%
+1 52.5% 42.8% 4.8%
+2 57.2% 38.2% 4.5%
+3 61.8% 34.0% 4.2%
+4 66.0% 30.0% 4.0%
+5 70.0% 26.2% 3.8%
+6 73.8% 22.8% 3.5%
+7 77.2% 19.5% 3.2%
+8 80.5% 16.5% 3.0%
+9 83.5% 13.8% 2.8%
+10 86.2% 11.2% 2.5%

Opposed skill checks

In D&D (3rd edition), standard skill checks are easy. You roll a d20, add your bonuses, and compare it to the DC (difficulty class). Your chance of failure is 5% per point of difference between your skill bonuses and the DC. It's a simple linear progression.

Opposed checks are more difficult to figure the odds. In an opposed check, you compare your roll against someone else's. For example, a thief trying to sneak past a guard would roll his Move Silently versus the guard's Listen. Or two players might compare their Ride skill rolls to determine who wins a horse race.


Chance of rolling each attribute
Stat Modifier Probability
(3d6)
Probability
(4d6 drop 1)
3 -4 0.5% 0.1%
4 -3 1.4% 0.3%
5 -3 2.8% 0.8%
6 -2 4.6% 1.6%
7 -2 6.9% 2.9%
8 -1 9.7% 4.8%
9 -1 11.6% 7.0%
10 +0 12.5% 9.4%
11 +0 12.5% 11.4%
12 +1 11.6% 12.9%
13 +1 9.7% 13.3%
14 +2 6.9% 12.3%
15 +2 4.6% 10.1%
16 +3 2.8% 7.3%
17 +3 1.4% 4.2%
18 +4 0.5% 1.6%

Attributes

The "traditional" method of rolling up attributes is to roll 3d6 and add them together, producing a nice traditional bell curve with 10 and 11 being in the center; the average result is 10½. Of course, heros are supposed to be extraordinary, so everyone used house rules to improve the average stat. D&D (3rd edition) has tried to codify the rules to "roll 4d6 and add the highest three dice together". This bends the bell curve a bit; the average moves up to 12¼ and 18s are (slightly) more likely.

How to read this table:


Chance of generating a character, based on total modifiers
Modifier Chance At least
+1 4.90% 100.00%
+2 7.00% 95.10%
+3 9.50% 88.10%
+4 11.20% 78.60%
+5 12.50% 67.40%
+6 12.60% 54.90%
+7 11.60% 42.30%
+8 10.00% 30.70%
+9 7.70% 20.70%
+10 5.40% 13.00%
+11 3.60% 7.60%
+12 2.10% 4.00%
+13 1.10% 1.90%
+14 0.50% 0.80%
+15 0.20% 0.30%
+16 0.10% 0.10%

Generating a character

This section assumes that you use the standard D&D rules for generating a character:

  1. Roll 4d6, remove the lowest die. Do this 6 times.
  2. If none of the stats are higher than a 13, start over at step 1.
  3. If the total modifiers are 0 or less, start over at step 1.

This table was generated by having a computer roll up 100,000 characters using the above rules. Racial bonuses are not considered, but they will not change the total modifier except in the case of the half-orc.

How to read this table:


Highest attribute
Highest
attribute
Equal At least
14 13% 100%
15 23% 87%
16 30% 64%
17 23% 34%
18 11% 11%

Single highest attribute

Everyone wants to have at least one 18, but sometimes the dice don't cooperate. Using the same dataset as the before, and thus the same (standard) rules, this chart shows the relative chance of getting each high stat.

So, we can see that 100% of characters will have at least one 14 or higher (since you re-roll characters who don't meet that minimum), but only 11% will have an 18. Of course, this is before any racial modifiers, which will skew the results by quite a bit.

How to read this table:


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