- 05 Jun '09 22:00The game Dungeons and Dragons uses various kinds of dice. Most are in the shape of the regular polyhedrons (cubes, octahedrons, etc).

When three cube shaped six sided dice are rolled and the total taken, a number from 3 to 18 is found. The distribution of the possible results is a bell shaped curve with an expectation value and peak at 10.5. Thus, the average roll of 3d6 (three six sided dice) is ten and a half (sometimes written as 10-11).

If 1d4, 1d6 and 1d8 were rolled, you'd likewise get a value from 3 to 18. The expectation value is again 10.5.

Is the distribution different? I couldn't figure that part out in my head. - 05 Jun '09 22:54

Yes.*Originally posted by AThousandYoung***The game Dungeons and Dragons uses various kinds of dice. Most are in the shape of the regular polyhedrons (cubes, octahedrons, etc).**

When three cube shaped six sided dice are rolled and the total taken, a number from 3 to 18 is found. The distribution of the possible results is a bell shaped curve with an expectation value and peak at 10.5. Thu ...[text shortened]... e is again 10.5.

Is the distribution different? I couldn't figure that part out in my head.

3d6 gives you 3 ways to get a 17 (566, 656, 665), whereas the 4-6-8 combo only gives one way (467). - 06 Jun '09 01:32Distribution for 4-6-8 dice:

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distribution for 6-6-6 dice:

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* - 06 Jun '09 04:50as forkedknight showed so nicely in a visual format above, 6-6-6 is "more heavily weighted" towards the center. there are the same number of combinations to achieve the extremal values, but a larger number of total permutations (6*6*6 = 216 dice rolls, vs. 4*6*8 = 192 dice rolls). and it stands to reason that since the possible values of the dice are closer together, that those additional permutations will be the "middle" values of the dice rolls.

but an actual analysis of all the possibilities is much clearer and provides real evidence, rather than just a "rule of thumb" or intuitive reasoning, so kudos to forkedknight for drawing it out for us!