Character Stat Probabilities (3.5e Other)

Given that a standard 2nd character stat roll is 3d6, what is the probability of rolling a value of j (such that j is between 3 and 18, obviously)?

It is trivial to compute these probabilities manually or by computer; we are looking for a general explicit formula for this probability. As it turns out, the probability can be expressed as the coefficients of this polynomial,

<math>\tfrac{1}{6^3}(x+x^2+x^3+x^4+x^5+x^6)^3</math>

In this case, the coefficient of x_n will be the probability of rolling an n with a 3d6 roll.

While this gives a hint at possible approach, this is to specific and we now consider the general problem; what is the probability of rolling a j with a ndk roll? As before, the coefficients of this polynomial give the answers,

<math>\tfrac{1}{k^n}(x+x^2+x^3+...+x^k)^n</math>

however we need to find an explicit form of these coefficients. This might be done by using the binomial theorem repeatedly, such as for k=3,

<math>\tfrac{1}{3^n}(x+x^2+x^3)^n=x^n(1+x(1+x))^n</math>

However, it proves quite cumbersome to use this method for more complex polynomials. It is better just to expand the polynomial out using a program such as Mathematica. Doing this, results in this polynomial;

(x + x² + x³ + x⁴ + x⁵ + x⁶)³ = x³ + 3x⁴ + 6x⁵ + 10x⁶ + 15x⁷ + 21x⁸ + 25x⁹ + 27x¹⁰ + 27x¹¹ + 25x¹² + 21x¹³ + 15x¹⁴ + 10x¹⁵ + 6x¹⁶ + 3x¹⁷ + x¹⁸

The coefficient of the nth power of x represents the number of ways to get n by rolling three six-sides dice and adding them together. Of course, multiplying the polynomial by 6^-3 will give the probability of rolling an n.

A harder problem I faced is the more standard "4d6 drop the lowest" roll. Computing the probability algebraically seems really difficult; luckily, someone else took interest and computed the probabilities with a program. This list gives the probability of rolling a 3 to 18 and the number of ways it can be rolled out of a total number of 1296 ways.

• 3 1 0.08%
• 4 4 0.31%
• 5 10 0.77%
• 6 21 1.62%
• 7 38 2.93%
• 8 62 4.78%
• 9 91 7.02%
• 10 122 9.41%
• 11 148 11.42%
• 12 167 12.89%
• 13 172 13.27%
• 14 160 12.35%
• 15 131 10.11%
• 16 94 7.25%
• 17 54 4.17%
• 18 21 1.62%

Total 1296 100.00%

Character percentile ranking

Now that I have these probabilities of a standard 4d6 roll, I can now attempt to settle the problem of making a "percentile ranking" of your character, that is, the see what the probability is of rolling a "worse" character.

To define what a character is worse or better than a given character, it is best to consider the total sum of your stat modifiers. Also, I define fractional stat modifiers for stats that are "in between" to integer stat modifiers. For example a 17 has a stat modifier of 3.5 and a 8 has a stat mod of -1.5 . Here is a list of all the stat modifiers,

• Stat Modifier
• 3 -4
• 4 -3.5
• 5 -3
• 6 -2.5
• 7 -2
• 8 -1.5
• 9 -1
• 10 -.5
• 11 .5
• 12 1
• 13 1.5
• 14 2
• 15 2.5
• 16 3
• 17 3.5
• 18 4

The problem is then, what is the probability of rolling a character (with six stats using the standard 4d6 roll), such that the sum of the stat mods is less than yours? To solve it, I use the same general idea as the previous problems and create a polynomial which I then multiply by itself several times. In this case, the polynomial is:

Where p(k) is the probability of rolling and k with a 4d6 dice roll given by the aforementioned table.

Then the coefficients of xⁿ of the polynomial p(x)⁶ will represent the probability of rolling a character with a stat mod total of n, and the sum of the coefficients before n will be the probability of rolling a "worse" character. Obviously, this resulting polynomial is huge, but luckily Mathematica handles this stuff with ease. Here is the resulting percentile ranking.

• -24 0
• -23.5 2.11043 x 10^(-19)
• -23 5.27606 x 10^(-18)
• -22.5 6.85888 x 10^(-17)
• -22 6.18566 x 10^(-16)
• -21.5 4.35149 x 10^(-15)
• -21 2.54422 x 10^(-14)
• -20.5 1.28603 x 10^(-13)
• -20 5.77152 x 10^(-13)
• -19.5 2.25356 x 10^(-12)
• -19 8.73246 x 10^(-12)
• -18.5 3.0176 x 10^(-11)
• -18 9.74638 x 10^(-11)
• -17.5 2.96276 x 10^(-10)
• -17 8.52027 x 10^(-10)
• -16.5 2.32815 x 10^(-9)
• -16 6.06693 x 10^(-9)
• -15.5 1.5125 x 10^(-8)
• -15 3.61723 x 10^(-8)
• -14.5 8.31888 x 10^(-8)
• -14 1.84376 x 10^(-7)
• -13.5 3.94598 x 10^(-7)
• -13 8.16979 x 10^(-7)
• -12.5 0.00000163913
• -12 0.00000319201
• -11.5 0.00000604266
• -11 0.0000111362
• -10.5 0.0000200077
• -10 0.0000350893
• -9.5 0.0000601463
• -9 0.000100878
• -8.5 0.00016573
• -8 0.000266956
• -7.5 0.000421981
• -7 0.000655094
• -6.5 0.000999495
• -6 0.0014997
• -5.5 0.0022142
• -5 0.00321878
• -4.5 0.00460897
• -4 0.00650383
• -3.5 0.00904829
• -3 0.0124154
• -2.5 0.0168074
• -2 0.0224557
• -1.5 0.0296188
• -1 0.0385777
• -.5 0.0496304
• 0 0.0630819
• .5 0.0792355
• 1 0.0983711
• 1.5 0.120737
• 2 0.146528
• 2.5 0.175871
• 3 0.208805
• 3.5 0.24527
• 4 0.285094
• 4.5 0.327985
• 5 0.373534
• 5.5 0.42122
• 6 0.470424
• 6.5 0.52045
• 7 0.570551
• 7.5 0.619957
• 8 0.667912
• 8.5 0.713706
• 9 0.756704
• 9.5 0.796382
• 10 0.832338
• 10.5 0.864317
• 11 0.892206
• 11.5 0.916035
• 12 0.935959
• 12.5 0.952245
• 13 0.965238
• 13.5 0.975342
• 14 0.982987
• 14.5 0.988606
• 15 0.992608
• 15.5 0.995365
• 16 0.997199
• 16.5 0.998372
• 17 0.999094
• 17.5 0.999518
• 18 0.999756
• 18.5 0.999883
• 19 0.999947
• 19.5 0.999978
• 20 0.999991
• 20.5 0.999997
• 21 0.99999910
• 21.5 0.999999703
• 22 0.99999992534
• 22.5 0.999999984338
• 23 0.9999999974212
• 23.5 0.99999999970263
• 24 0.99999999998189

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To some players, this percentile ranking may seem a little off, in particular, a little underestimating of your character. For example, if you characters stats are {8,8,17,17,18,10}, you may think that you have an outstanding set of stats, when in fact you do not. The stat mod total of this particular set of stats is only 8, which equates to a 33% chance of rolling a character with a higher stat mod. What gives this illusion of superiority is the presence of the "high stats"; 17, 17 and 18. The mind seems to focus on these stats only, and while these are high stats, they are counterbalanced by the two 8's, which are actually far below average in the 4d6 roll.

To give different idea of character percentile ranking, or what is a "good" character, one may simply focus on the "high stats". This makes sense in Dnd since it is generally better in-game to have a specialized character versus a non-specialized one. Lets assume that a good notion of high stats would be stats that have a modifier of more than 3, say. Then 16 17 and 18 are the "high stats" that players go nuts over. We can now can consider the probability of rolling n number of high stats in a character stat set (six stats);

p(1)=0.38907

p(2)=0.145858

p(3)=0.029163

p(4)=0.00327986

p(5)=0.000196734

p(6)=4.91688x10^-6

As we can see, rolling 1 stat 16 or higher is quite common at a 38% chance, rolling 2 stats not very common, rolling 3 and 4 stats very rare, and rolling 5 and 6 (all stats) higher than 16 practically impossible. Under this line of thinking, the aformentioned stat set has 3 "high" stats greater or equal to 16 is actually pretty rare with only a 2.9% chance. However, having 3 high stats and 3 5's would obviously not be a "good" character, so I guess a good measure as to how good your stat set is, is to take a balance of these two "rankings".

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