Dice, Cards, and Probability part 4 More than one dice

In this Dice, Cards, and Probability Series

In Part 1, I looked at probability, notation, number of occurrences.

In Part 2, I looked at linear die/dice (single) results

In Part 3, I looked at the d12, rolling over, rolling under, modifiers, target numbers, difficulty, proficiency, and values

More than one dice

Comparing the 1d12 flat probability to multiple dice, a dramatic shift in shape and probability results from adding more dice.

Looking at max 12 combinations

Following along my own interest, I will consider three dice combinations that can fit in with the 1d12 for comparison. These are 2d6, 3d4, and 4d3. I do this for comparison purpose. In a similar manner, other dice combinations can be looked at with the same principles and influence of using multiple number of a certain dice.

2d6

2d6 is an extremely popular system at the moment.

Its results form a pyramid shaped probability.

3d4

This probability is more round.

4d3

This is the basis for Fate and Fudge dice systems. This is a steep bell shape.

Fate/Fudge

This is a popular cinematic system, the current edition is the third edition. It is based on 4dF dice which have a -, +, and 0 sides meaning -1, +1, and 0. Because of rolling four dice, the results are a curved probability of -4 to +4 with the center 0 being the most frequent result.

Comparing the four dice that max at 12 result

1d12 is very flat.
2d6 is a sharp pyramid shape
3d4 is rounded
4d3 is a steep bell curve shape

Of course, the more the dice the higher the initial number, effectively squeezing the available results narrower.

At the ends, the probability goes down from 8.33, to 2.78, to 1.56, to 1.23.
In the middle, the probability goes up from 8.33, to 16.67, to 18.75, to 23.46.

Summary

A linear system is fine with a plain 1d12 and 2d6.
Curvilinear systems may want distributed results that 3d4 and 4d3 (4dF) gives a game.