Wednesday, October 28, 2009

Hackmaster / Exploding Dice Analysis


Back when reviewing Hackmaster I wondered about the penetration (exploding) dice. I worried that a d4 would be better than d6 as the d4 was more likely to roll it's max value and explode than the d6. Being a programmer dude I wrote code and ran some tests.

The results of "throwing" each type of die 100,000 times.
die   avg    max
d4p 3.01 24
d6p 3.99 37
d8p 5.00 49
d12p 7.00 55
As you can see, exploding only increases the avg by about .5 and max values ramp up nicely. If you roll 100,000 times you can get some impressive max rolls. But, super high explosions are rare. Here are the details on d4, d6, and d12 (chosen cause of this) rolled "only" 12,000 times each. Results are value rolled, number of times it was rolled, % chance of rolling that value, % chance of rolling that value or higher.
d4p count chance cumulative
1 2954 24.62% 100.00%
2 3060 25.50% 75.38%
3 2991 24.93% 49.88%
4 756 6.30% 24.96%
5 796 6.63% 18.66%
6 740 6.17% 12.02%
7 177 1.47% 5.86%
8 184 1.53% 4.38%
9 181 1.51% 2.85%
10 43 0.36% 1.34%
11 43 0.36% 0.98%
12 42 0.35% 0.62%
13 11 0.09% 0.27%
14 3 0.03% 0.18%
15 10 0.08% 0.16%
16 5 0.04% 0.07%
17 not rolled
18 3 0.03% 0.03%
19 1 0.01% 0.01%
20 not rolled
21 1 0.01% -0.00%

d6p count chance cumulative
1 1951 16.26% 100.00%
2 2064 17.20% 83.74%
3 1922 16.02% 66.54%
4 1993 16.61% 50.53%
5 1992 16.60% 33.92%
6 334 2.78% 17.32%
7 347 2.89% 14.53%
8 331 2.76% 11.64%
9 370 3.08% 8.88%
10 322 2.68% 5.80%
11 60 0.50% 3.12%
12 68 0.57% 2.62%
13 43 0.36% 2.05%
14 75 0.62% 1.69%
15 64 0.53% 1.07%
16 7 0.06% 0.53%
17 11 0.09% 0.48%
18 9 0.07% 0.38%
19 12 0.10% 0.31%
20 16 0.13% 0.21%
21 not rolled
22 not rolled
23 4 0.03% 0.08%
24 2 0.02% 0.04%
25 1 0.01% 0.03%
26 not rolled
27 not rolled
28 2 0.02% 0.02%
29 1 0.01% 0.00%

d12p count chance cumulative
1 968 8.07% 100.00%
2 978 8.15% 91.93%
3 991 8.26% 83.78%
4 1004 8.37% 75.53%
5 987 8.22% 67.16%
6 996 8.30% 58.93%
7 980 8.17% 50.63%
8 982 8.18% 42.47%
9 1006 8.38% 34.28%
10 1025 8.54% 25.90%
11 1057 8.81% 17.36%
12 87 0.73% 8.55%
13 78 0.65% 7.83%
14 74 0.62% 7.18%
15 80 0.67% 6.56%
16 88 0.73% 5.89%
17 99 0.83% 5.16%
18 82 0.68% 4.33%
19 93 0.78% 3.65%
20 95 0.79% 2.87%
21 83 0.69% 2.08%
22 84 0.70% 1.39%
23 6 0.05% 0.69%
24 8 0.07% 0.64%
25 7 0.06% 0.57%
26 5 0.04% 0.52%
27 10 0.08% 0.47%
28 7 0.06% 0.39%
29 8 0.07% 0.33%
30 5 0.04% 0.27%
31 8 0.07% 0.22%
32 7 0.06% 0.16%
33 5 0.04% 0.10%
34 not rolled
35 3 0.03% 0.06%
36 not rolled
37 2 0.02% 0.03%
38 not rolled
39 not rolled
40 not rolled
41 not rolled
42 not rolled
43 not rolled
44 1 0.01% 0.02%
45 not rolled
46 not rolled
47 1 0.01% 0.01%
48 not rolled
49 1 0.01% -0.00%
5% chance for 17+ damage is a little high (2d12 avg=13). Seems not too out of whack for "critical hit" system. Still might change 2-hand damage to d10p.

In my Post on Weapon Damage a comment was made that exploding dice combined with my house rule, pure fighters get to roll damage twice and take the higher result, is too much of a damage escalation. An unmentioned tweak is fighters get to take the higher of the initial non-exploded roll, they don't roll exploded dice twice. Mostly cause I don't want to deal with insane amount of dice rolling / tracking.

So, modifying my program... Here are 2d10 and 2d12 rolled 12000 times, highest initial roll taken, and then exploded as appropriate.
d10p count chance cumulative
1 131 1.09% 100.00%
2 351 2.93% 98.91%
3 565 4.71% 95.98%
4 867 7.22% 91.28%
5 1082 9.02% 84.05%
6 1278 10.65% 75.03%
7 1600 13.33% 64.38%
8 1763 14.69% 51.05%
9 2072 17.27% 36.36%
10 232 1.93% 19.09%
11 241 2.01% 17.16%
12 233 1.94% 15.15%
13 218 1.82% 13.21%
14 213 1.77% 11.39%
15 243 2.02% 9.62%
16 226 1.88% 7.59%
17 207 1.73% 5.71%
18 234 1.95% 3.98%
19 26 0.22% 2.03%
20 23 0.19% 1.82%
21 24 0.20% 1.63%
22 13 0.11% 1.43%
23 28 0.23% 1.32%
24 27 0.22% 1.08%
25 33 0.27% 0.86%
26 19 0.16% 0.58%
27 23 0.19% 0.43%
28 5 0.04% 0.23%
29 2 0.02% 0.19%
30 1 0.01% 0.18%
31 6 0.05% 0.17%
32 2 0.02% 0.12%
33 3 0.03% 0.10%
34 2 0.02% 0.08%
35 1 0.01% 0.06%
36 7 0.06% 0.05%

d12p count chance cumulative
1 83 0.69% 100.00%
2 227 1.89% 99.31%
3 416 3.47% 97.42%
4 573 4.78% 93.95%
5 767 6.39% 89.17%
6 885 7.38% 82.78%
7 1067 8.89% 75.41%
8 1291 10.76% 66.52%
9 1366 11.38% 55.76%
10 1590 13.25% 44.38%
11 1813 15.11% 31.12%
12 148 1.23% 16.02%
13 149 1.24% 14.78%
14 182 1.52% 13.54%
15 163 1.36% 12.02%
16 141 1.18% 10.67%
17 159 1.32% 9.49%
18 153 1.27% 8.17%
19 181 1.51% 6.89%
20 154 1.28% 5.38%
21 148 1.23% 4.10%
22 182 1.52% 2.87%
23 13 0.11% 1.35%
24 13 0.11% 1.24%
25 20 0.17% 1.13%
26 14 0.12% 0.97%
27 11 0.09% 0.85%
28 19 0.16% 0.76%
29 7 0.06% 0.60%
30 10 0.08% 0.54%
31 10 0.08% 0.46%
32 18 0.15% 0.37%
33 19 0.16% 0.22%
34 1 0.01% 0.07%
35 1 0.01% 0.06%
36 1 0.01% 0.05%
37 not rolled
38 not rolled
39 not rolled
40 2 0.02% 0.04%
41 1 0.01% 0.02%
42 2 0.02% 0.02%
43 1 0.01% -0.00%
Chance of exploding is 2x normal what I'd expected. Did not realize "rolling twice taking best" would produce such high results. 55% chance of 9 or higher on d12.


For completeness here's d6p as rolled by fighter.
d6p  count chance cumulative
1 328 2.73% 100.00%
2 1016 8.47% 97.27%
3 1727 14.39% 88.80%
4 2322 19.35% 74.41%
5 2937 24.47% 55.06%
6 620 5.17% 30.58%
7 589 4.91% 25.42%
8 580 4.83% 20.51%
9 632 5.27% 15.68%
10 612 5.10% 10.41%
11 104 0.87% 5.31%
12 101 0.84% 4.44%
13 104 0.87% 3.60%
14 107 0.89% 2.73%
15 102 0.85% 1.84%
16 18 0.15% 0.99%
17 21 0.18% 0.84%
18 18 0.15% 0.67%
19 23 0.19% 0.52%
20 22 0.18% 0.33%
21 1 0.01% 0.14%
22 3 0.03% 0.13%
23 2 0.02% 0.11%
24 2 0.02% 0.09%
25 5 0.04% 0.08%
26 not rolled
27 not rolled
28 not rolled
29 1 0.01% 0.03%
30 1 0.01% 0.03%
31 1 0.01% 0.02%
32 not rolled
33 not rolled
34 not rolled
35 2 0.02% 0.01%

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