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### Qualifying for character classes in Advanced Dungeons & Dragons

When rolling ability scores in AD&D, particular minimum values must be achieved in order to qualify for each race and class. We will show by an example how one may calculate the probability of meeting these requirements; generalizing the method will be left for the interested reader.

Consider the Human Fighter, which needs the following minimum scores: Strength 9, Intelligence 3, Wisdom 6, Dexterity 6, Constitution 7, Charisma 6.

Let $P(a)$ denote the probability of rolling a single score of exactly $a$ and let $P(a,b)$ denote the probability of rolling a single score of at least $a$ and at most $b$.

If the scores must be rolled in order, then the probability of qualifying for the class is found by multiplying together the probabilities of meeting or exceeding each minimum value: $$P(3,18)P(6,18)^3P(7,18)P(9,18).$$

For 3d6 that works out to about 58.3 percent.

If the rolling order doesn't matter, then the problem is more complex. Observe that we have the following requirements:

1. Six scores must be at least 3;
2. of those, at least five must be 6 or greater;
3. of those, at least two must be 7 or greater;
4. and of those, at least one must be nine or greater.

Let $i$ be the number of scores of nine or higher. If $j$ is the maximum number of scores which are at least 7 but less than 9, then for any $i$ the number of such scores will be $j-i$. Similarly, if $k$ is the maximum number of scores which are at least 6 but less than 7, then for any $j$ the number of such scores will be $k-j$. That leaves $6-k$ scores which are at least 3 but less than 6.

The probability of rolling $i$ scores of probability $P(9,18)$, $j-i$ scores of probability $P(7,8)$, $k-j$ scores of probability $P(6)$, and $6-k$ scores of probability $P(3,5)$ has a multinomial distribution: $$\frac{6!}{i!(j-i)!(k-j)!(6-k)!}P(9,18)^iP(7,8)^{j-i}P(6)^{k-j}P(3,5)^{6-k}.\qquad\textrm{(1)}$$

Now from the requirements above, $k$ can vary from 5 to 6, so it follows that $j$ may vary from 2 to $k$, and hence $i$ can vary from 1 to $j$. Summing $\textrm{(1)}$ for all these variations, we find the total probability of qualifying for the Fighter class: $$\sum_{k=5}^6\sum_{j=2}^k\sum_{i=1}^j\frac{6!}{i!(j-i)!(k-j)!(6-k)!}P(9,18)^iP(7,8)^{j-i}P(6)^{k-j}P(3,5)^{6-k}.$$

For 3d6 that works out to about 97.14 percent.


Minimum scores    Unordered    Ordered
Character StrIntWisDexConCha 3d6 3/4d6 3d6 3/4d6
Cleric66936697.140%99.805%61.280%85.434%
Druid6612615630.868%73.455%2.873%13.622%
Fighter93667697.140%99.805%58.305%84.033%
Ranger13131461463.256%29.351%0.161%2.927%
Magic-user39666697.140%99.805%61.280%85.434%
Illusionist615616367.562%35.744%0.372%2.915%
Thief66396697.140%99.805%61.280%85.434%
Assassin12116126358.571%93.483%6.395%27.134%
Monk1561561167.625%39.576%0.372%3.781%
Bard1512151510150.027%1.581%0.002%0.146%
Half-elf Cleric66966675.236%93.253%58.443%84.445%
Half-orc Cleric569312892.909%99.643%21.788%51.230%
Half-elf Druid6612615630.868%73.455%2.873%13.622%
Dwarf Fighter936611795.464%99.763%30.568%62.123%
Elf Fighter98668873.219%93.089%39.643%73.306%
Gnome Fighter97668675.222%93.253%48.858%79.235%
Half-elf Fighter94667694.949%99.368%58.035%83.968%
Halfling Fighter1066710673.678%93.193%30.747%63.877%
Half-orc Fighter836612891.247%99.481%23.950%53.550%
Half-elf Ranger13131461463.256%29.351%0.161%2.927%
Elf Magic-user39667897.089%99.804%51.230%80.163%
Half-elf Magic-user39666697.140%99.805%61.280%85.434%
Gnome Illusionist615616866.269%33.908%0.312%2.749%
Dwarf Thief863911794.713%99.731%26.858%59.262%
Elf Thief68387896.171%99.771%50.920%80.558%
Gnome Thief67398697.089%99.804%51.230%80.163%
Half-elf Thief66396697.140%99.805%61.280%85.434%
Halfling Thief763810696.883%99.802%43.225%73.874%
Half-orc Thief563912892.909%99.643%21.788%51.230%
Dwarf Assassin121161211432.381%79.058%3.337%20.044%
Elf Assassin12116117851.151%88.806%6.798%29.827%
Gnome Assassin12116128357.852%93.361%5.619%25.884%
Half-elf Assassin12116126358.571%93.483%6.395%27.134%
Half-orc Assassin111161212531.003%78.002%3.291%19.982%
Half-elf Bard1512151510150.027%1.581%0.002%0.146%
Half-elf Cleric/Fighter96967675.038%93.249%43.189%75.215%
Half-orc Cleric/Fighter869612869.795%92.831%17.741%47.931%
Half-elf Cleric/Fighter/Magic-user99967673.268%93.152%33.545%68.110%
Half-elf Cleric/Ranger13131461463.256%29.351%0.161%2.927%
Half-elf Cleric/Magic-user69966675.042%93.250%45.393%76.468%
Half-orc Cleric/Thief569912879.540%97.012%16.139%45.854%
Half-orc Cleric/Assassin111191212817.116%59.696%2.182%17.127%
Elf Fighter/Magic-user99668873.137%93.087%35.044%69.587%
Half-elf Fighter/Magic-user99667675.038%93.249%43.189%75.215%
Gnome Fighter/Illusionist915616866.210%33.885%0.242%2.489%
Dwarf Fighter/Thief966911772.650%93.127%22.643%55.604%
Elf Fighter/Thief98688863.313%90.632%34.832%69.930%
Gnome Fighter/Thief97698674.781%93.237%37.948%71.751%
Half-elf Fighter/Thief96697675.038%93.249%43.189%75.215%
Halfling Fighter/Thief1066810673.575%93.189%28.394%61.950%
Half-orc Fighter/Thief866912869.795%92.831%17.741%47.931%
Elf Fighter/Magic-user/Thief99688863.286%90.631%30.791%66.382%
Half-elf Fighter/Magic-user/Thief99697673.268%93.152%33.545%68.110%
Elf Magic-user/Thief69687874.924%93.240%42.928%75.586%
Half-elf Magic-user/Thief69696675.042%93.250%45.393%76.468%
Gnome Illusionist/Thief615616866.269%33.908%0.312%2.749%



See part two to find probabilities for multiple attempts.