Many people wonder about the exact probabilities of any random activity in which they are involved, and role playing games are no exception. Some games (e.g. d20) have simple, flat probability curves, while others use strange, dificult to mathematically evaluate systems (Legend of the Five Rings, Deadlands). My system of choice, GURPS, uses a 3d6 bell curve, and I'm going to evaluate the possibilities of each result, along with an explanaiton on how I came to these results.
First at all, a fair d6 does have a flat probability curve. Each number has the exact same probability. When you roll 2d6 and add them together, you get a pyramid like distribution:
| Target Number | Nº of combinations | Combinations |
| 2 | 1 | (1,1) |
| 3 | 2 | (1,2) (2,1) |
| 4 | 3 | (1,3) (2,2) (3,1) |
| 5 | 5 | (1,4) (2,3) (3,2) (4,1) |
| 6 | 5 | (1,5) (2,4) (3,3) (4,2) (5,1) |
| 7 | 6 | (1,6) (2,5) (3,4) (4,3) (5,2) (6,1) |
| 8 | 5 | (2,6) (3,5) (4,4) (5,3) (6,2) |
| 9 | 4 | (3,6) (4,5) (5,4) (6,3) |
| 10 | 3 | (4,6) (5,5) (6,4) |
| 11 | 2 | (5,6) (6,5) |
| 12 | 1 | (6,6) |
This is relatively easy, because the total number of combinations is quite small (36). However, for 3d6, the number of combinations is 216! There is, however, a trick:
In order to count the number of possibilities, you must remember that for each target number above, you've got 6 new possibilities, one for each result on the third die. It's obvious to see that the number of ways to get 4, for example is 3, ([1,1],2) ([1,2],1) ([2,1],1). In fact, the number of possibilities is 1+2.
Using this, we quicly arrive at the following table:
| Target Number | Number of possibilities | |
| 3 | 1 | =1 |
| 4 | 3 | =1+2 |
| 5 | 6 | =1+2+3 |
| 6 | 10 | =1+2+3+4 |
| 7 | 15 | =1+2+3+4+5 |
| 8 | 21 | =1+2+3+4+5+6 |
| 9 | 25 | =2+3+4+5+6+5 |
| 10 | 27 | =3+4+5+6+5+4 |
| 11 | 27 | =4+5+6+5+4+3 |
| 12 | 25 | =5+6+5+4+3+2 |
| 13 | 21 | =6+5+4+3+2+1 |
| 14 | 15 | =5+4+3+2+1 |
| 15 | 10 | =4+3+2+1 |
| 16 | 6 | =3+2+1 |
| 17 | 3 | =2+1 |
| 18 | 1 | =1 |
This table show the distribution of possibilities. Dividing the number of possibilities by 216, we get the probabilities of each number. This is what we will call the probability density functions, fx(X). If we add the probabilities up to a given number, we get the probability distribution function, Fx(X). we tabulate fx(X) and Fx(X) and we get:
| Target Number | fx(X) | Fx(X) |
| 3 | 1/216=0.46% | 1/216=0.46% |
| 4 | 3/216=1.39% | 4/216=1.85% |
| 5 | 6/216=2.78% | 10/216=4.63% |
| 6 | 10/216=4.63 | 20/216=9.26% |
| 7 | 15/216=6.94% | 35/216=16.20% |
| 8 | 21/216=9.72% | 56/216=25.93% |
| 9 | 25/216=11.57% | 81/216=37.50% |
| 10 | 27/216=12.50% | 108/216=50.00% |
| 11 | 27/216=12.50% | 135/216=62.50% |
| 12 | 25/216=11.57% | 160/216=74.07% |
| 13 | 21/216=9.72% | 181/216=83.80% |
| 14 | 15/216=6.94% | 196/216=90.74% |
| 15 | 10/216=4.63% | 206/216=95.37% |
| 16 | 6/216=2.78% | 212/216=98.15% |
| 17 | 3/216=1.39% | 215/216=99.54% |
| 18 | 1/216=0.46% | 216/216=100% |
Bellow are the graphics for fx(X) and Fx(X)
Now, this is for using three standard six sided dice. There is however, a nonstandard 3d6 configurations that gives the same probability distributions:
- 1, 2, 2, 3, 3, 4
- 1, 2, 3, 4, 5, 6
- 1, 3, 4, 5, 6, 8
The calculations done to find this are found in this page.
Now, we've explored the possibilities of rolling 3d6. There are, however, 2 alternate dice configurations that give results similar to the ones we get from 3d6, but that use only standard polyhedral dice: 1d4+1d6+1d8 and 2d4+1d10.
The following table shows the fx(X) for all 3 dice sets (3d6, 1d4+1d6+1d8 and 2d4+1d10)
For 2d4+1d10
| nº | Possibilities | fx(X) | Possibilities | Fx(X) |
| 3 | 1 | 0.625% | 1 | 0.625% |
| 4 | 3 | 1.875% | 4 | 2.5% |
| 5 | 6 | 3.75% | 10 | 6.25% |
| 6 | 10 | 6.25% | 20 | 12.5% |
| 7 | 13 | 8.125% | 33 | 20.625% |
| 8 | 15 | 9.375% | 48 | 30% |
| 9 | 16 | 10% | 64 | 40% |
| 10 | 16 | 10% | 80 | 50% |
| 11 | 16 | 10% | 96 | 60% |
| 12 | 16 | 10% | 112 | 70% |
| 13 | 15 | 9.375% | 127 | 79.375% |
| 14 | 13 | 8.125% | 140 | 87.5% |
| 15 | 10 | 6.25% | 150 | 93.75% |
| 16 | 6 | 3.75% | 156 | 97.5% |
| 17 | 3 | 1.875% | 159 | 99.375% |
| 18 | 1 | 0.625% | 160 | 100% |
For 1d4+1d6+1d8
| nº | Possibilities | fx(X) | Possibilities | Fx(X) |
| 3 | 1 | 0.521% | 1 | 0.52% |
| 4 | 3 | 1.563% | 4 | 2.08% |
| 5 | 6 | 3.125% | 10 | 5.21% |
| 6 | 10 | 5.208% | 20 | 10.42% |
| 7 | 14 | 7.292% | 34 | 17.71% |
| 8 | 18 | 9.375% | 52 | 27.08% |
| 9 | 21 | 10.938% | 73 | 38.02% |
| 10 | 23 | 11.979% | 96 | 50% |
| 11 | 23 | 11.979% | 119 | 61.98% |
| 12 | 21 | 10.938% | 140 | 72.92% |
| 13 | 18 | 9.375% | 158 | 82.29% |
| 14 | 14 | 7.292% | 172 | 89.58% |
| 15 | 10 | 5.208% | 182 | 94.79% |
| 16 | 6 | 3.125% | 188 | 97.92% |
| 17 | 3 | 1.563% | 191 | 99.48% |
| 18 | 1 | 0.521% | 192 | 100% |