6168 in-depth character profiles from comics, games, movies

Stock illustration about a pie chart

DC Heroes RPG – A short article on probabilities


Game system: DC Heroes Role-Playing Game

Context

This is a technical article for the DC Heroes RPG.

Mutants & Masterminds note: The DCH way to roll dice works just fine in M&M except at low PLs — see the discussion about using 2d10 rather than 1d20 in the Mastermind’s Manual.


Advertisement


Point of this article

The chief goal of this short article is to serve as a handy reference when setting game stats up. For instance :

  • To get a sense of the breakdown rate of a given Reliability Number (R#).
  • Ro gain a better sense of what it means exactly to say “OK, if I give Stupendous Man an OV of 08, then Nefarious Nelly needs a 14 to hit him”.

Mostly, I’m concerned about people assuming that the probabilities in DC Heroes are linear. As they would be if the game were using a d20 for rolls. It’s not the case – 2d10 form a normal distribution (or “bell curve” if you like cows, or “Gaussian distribution” if you like geniuses ).

Thankfully, the distribution formed by 2d10 is about the simplest practical normal distribution around. The matrix of possible results is regular, and computing the probabilities to roll under or over a given result can easily be done through quick mental calculation.

The trick is the exploding dice mechanics, where every double (save double ones, or ’snake eyes’) leads to an additive reroll. Obviously, this is not a trivial feature, probabilities-wise – 9% of rolls reemerging in the rightward tail of the distribution can be significant.


Advertisement


Straight 2d10 probabilities

As a quick intro, here are the probabilities for an unmodified 2d10 distribution. It’s simpler to grok , and as you’ll see it’s close to the actual distribution in practical terms.

Result Probabilities
2 20 1%
3 or less 19 or more 3%
4 or less 18 or more 6%
5 or less 17 or more 10%
6 or less 16 or more 15%
7 or less 15 or more 21%
8 or less 14 or more 28%
9 or less 13 or more 36%
10 or less 12 or more 45%
11 or less 11 or more 55%

Thus, you have, say, a 10% to chance to roll 5 or less. Note the difference from the results distribution of a d20, where one has a 25% chance to roll 5 or less. There is a 36% chance that any 2d10 roll will be 13 or above. And so on, and so forth.

(And yes, “11 or more” and “11 or less” add up to more to 100%, since 11 is part of both ranges. It’s in fact well above 100% since 11 is the mode – that is, the most common result.)


Working exploding dice in – basic version

The “explodes on a double” mechanic (that is, reroll and add the new roll to the original roll if you get a double) has two effects. It takes away 9 possible combinations, and transform each of those in a mini-normal distribution that adds to the normal curve to the right of its inception.

Since we’re living in such a modern world, let’s look at a picture of the distribution:

Distribution of dice rolls (2d10 exploding on doubles) in the DC Heroes RPG

This assumes one re-roll after a double, to keep things simple.


Chaining exploding dice

One of the Teeming Million of writeups.org readers kindly simulated a hundred million MEGS dice rolls to double-check. In this non-simplified version, the dice rolls keep going as long as it’s a double.

The impact on probabilities is about a 0.1% difference up to 20. But predictably, it plays a much more important role for high values.

Result Probabilities Score to beat Probabilities
3 or less 1.00% 3 99.00%
4 or less 3.00% 4 97.00%
5 or less 5.01% 5 94.99%
6 or less 9.01% 6 90.99%
7 or less 13.02% 7 86.98%
8 or less 19.04% 8 80.96%
9 or less 25.07% 9 74.93%
10 or less 33.13% 10 66.87%
11 or less 41.20% 11 58.80%
12 or less 51.32% 12 48.68%
13 or less 59.45% 13 40.55%
14 or less 67.65% 14 32.35%
15 or less 73.86% 15 26.14%
16 or less 80.17% 16 19.83%
17 or less 84.46% 17 15.54%
18 or less 88.84% 18 11.16%
19 or less 91.19% 19 8.81%
20 or less 93.64% 20 6.36%
21 or less 94.04% 21 5.96%
22 or less 94.53% 22 5.47%
23 or less 94.94% 23 5.06%
24 or less 95.46% 24 4.54%
25 or less 95.87% 25 4.13%
26 or less 96.37% 26 3.63%
27 or less 96.77% 27 3.23%
28 or less 97.24% 28 2.76%
29 or less 97.60% 29 2.40%
30 or less 98.01% 30 1.99%
31 or less 98.32% 31 1.68%
32 or less 98.65% 32 1.35%
33 or less 98.88% 33 1.12%
34 or less 99.12% 34 0.88%
35 or less 99.27% 35 0.73%
36 or less 99.43% 36 0.57%
37 or less 99.52% 37 0.48%
38 or less 99.62% 38 0.38%
39 or less 99.67% 39 0.33%
40 or less 99.72% 40 0.28%

Caveat !

The flat percentages above are IMO slightly deceptive.

It’s a distribution, so of course it simulates an infinite number of dice rolls, smoothing things over. That’s the point.

But there is still a 9% chance, which is certainly not negligible, to add a median 11 to your first roll. And the game mechanics, and in particular the Column Shifts, mean that it has a drastic impact.

This kind of “low probability, possibly high impact” event  is, within a 95% confidence interval, not well described by common statistical tools*. Here, the 91% of normally normally-distributed events are drowning the signal.

One should be careful to remember that:

  • People are spending Hero Points. Hence the “Rule of 15” in the rulebook, though as you can appreciate in the tables 15 is a fairly high number.
  • People are using Combat Manoeuvres. Which can be a pretty big deal, since 1 CS is significant. For instance, if Action Bob (DEX 07) is fighting Crazy Doug (DEX 06) and manages to Lay Back, Doug’s probabilities of hitting Bob went from roll-13-or-more to roll-15-or-more… which means his chances have dropped by about a third (from 40% to 26%).
  • Assuming two combatants, once every five Phases on average somebody’s rolling a double. Which might very well be a combat-ending event, with Column Shifts and whatnot.

* That was a joke.


Very low probability results

The 100,000,000 simulated dice rolls provided by our reader Vincent go up to 111 – presumably the highest MEGS dice roll you’ll ever hear about. For those with a morbid curiosity about very-low probability dice rolls in MEGS, here is an excerpt of his results table :

Target number Probability to roll that or more
40 0.275518%
45 0.143235%
50 0.059654%
55 0.023581%
60 0.010136%
65 0.004713%
70 0.001862%
75 0.000794%
80 0.000344%
85 0.000145%
90 0.000052%
95 0.000023%
100 0.000012%
105 0.000005%
110 0.000003%

The average result from this immense number of rolls was 12.066.

C# code

If somebody needs to simulate many rolls, Vincent’s code :
public int MEGS(int pretotal) {

int total = pretotal;
int rand1 = getRandomInt(1, 10);
int rand2 = getRandomInt(1, 10);

total = rand1 + rand2;
if (rand1 == rand2 && rand1 != 1) { total += MEGS(total); }
if (rand1 == rand2 && rand1 == 1) { total = 0; }

return total;

}


More numbers from our contributor Adrian

The following also approximates the probability distribution of the exploding 2D10 system. It was created by combining the individual probabilities of 15,183 rolls.

It started by looking at the distribution of the 90 ways of rolling two ten sided dice without rolling doubles that lead to a longer chain of rolls. It then looked at the series of additional combinations that were possible with each successive series created by rolling doubles.

Every “chain” of rolls terminates with either a set of non-doubles or with double ones. A chain of one has no doubles. A chain of two has one set of doubles and one terminating set. Here is a list with the number of combinations used for each chain.

Chain Combos
1 91
2 297
3 441
4 585
5 729
6 873
7 1,017
8 1,161
9 1,305
10 1,449
11 1,593
12 1,737
13 1,881
14 2,025
Total 15,183

The combined probability of rolling double ones is 1.09890…% with 109890 repeating.

And here is the table of probabilities. If that’s too many numbers for you, just scroll past to the next one.

Value Probability At or below ~rarity

2 0.010989010989011 0.010989010989011 91

3 0.02 0.030989010989011 32.2695035460993

4 0.02 0.050989010989011 19.6120689655172

5 0.04 0.090989010989011 10.9903381642512

6 0.04 0.130989010989011 7.63422818791946

7 0.0602 0.191189010989011 5.23042613603706

8 0.0602 0.251389010989011 3.9778986204123

9 0.0806 0.331989010989011 3.0121478931515

10 0.0806 0.412589010989011 2.42371942384727

11 0.101202 0.513791010989011 2.0567287372332

12 0.081202 0.594993010989011 2.4690931937791

13 0.082008 0.677001010989011 3.09598492262147

14 0.062008 0.739009010989011 3.83154990825333

15 0.06302002 0.802029030989011 5.05124566998756

16 0.04282002 0.844849050989011 6.44533601872568

17 0.0438401 0.888689150989011 8.98385026154347

18 0.0234401 0.912129250989011 11.3803513826306

19 0.0244703002 0.936599551189011 15.7727590065052

20 0.0038683002 0.940467851389011 16.7976467057231

21 0.0049087012 0.945376552589011 18.3071564940959

22 0.0041027012 0.949479253789011 19.7938485671552

23 0.005153404202 0.954632657991011 22.0422875953777

24 0.004141384202 0.958774042193011 24.2565619622903

25 0.005002491214 0.963776533407011 27.60641357814

26 0.003982411214 0.967758944621011 31.0163544041947

27 0.00465402525602 0.972412969877031 36.2489182613169

28 0.00362382505602 0.976036794933051 41.730644845135

29 0.00410205016816 0.980138845101211 50.3495393443087

30 0.00306164916816 0.983200494269371 59.5255608131837

31 0.00334259042072 0.986543084690091 74.3112352994926

32 0.00229188741872 0.988834972108811 89.5653830644857

33 0.002371590922402 0.991206563031213 113.721176776449

34 0.001510483910402 0.992717046941615 137.306940190784

35 0.001584936826609 0.994301983768224 175.49967555783

36 0.000913322784589 0.995215306552813 208.99980553359

37 0.000978452535833 0.996193759088646 262.726407310932

38 0.000500227423693 0.996693986512339 302.479104738144

39 0.000555900904099 0.997249887416438 363.621477163207

40 0.000274959651539 0.997524847067977 404.015439636923

41 0.000320982427457 0.997845829495434 464.215807374806

42 0.000241278923776 0.99808710841921 522.768781065476

43 0.000277454424498 0.998364562843709 611.457307395142

44 0.000203001508291 0.998567564352 698.111640404924

45 0.000229130206753 0.998796694558753 831.04419353692

46 0.000164000455509 0.998960695014262 962.181471004471

47 0.000181879838858 0.99914257485312 1166.28256546752

48 0.000126206358452 0.999268781211572 1367.57973923136

49 0.000137630888365 0.999406412099937 1684.67045890529

50 9.16081124467899E-05 0.999498020212384 1992.11208233803

51 9.84291850723E-05 0.999596449397456 2478.00398189433

52 6.22536843497089E-05 0.999658703081806 2930.00008699385

53 6.63795897487512E-05 0.999725082671554 3637.45714267741

54 4.02508912869748E-05 0.999765333562841 4261.36780405654

55 4.36479723754562E-05 0.999808981535217 5235.09599522514

56 2.57685890264539E-05 0.999834750124243 6051.44176612179

57 2.84624000068537E-05 0.99986321252425 7310.61081812432

58 1.70378700934336E-05 0.999880250394344 8350.75818846629

59 1.91143543722781E-05 0.999899364748716 9936.87586845524

60 1.22932817467679E-05 0.999911658030463 11319.6480137188

61 1.38399693313555E-05 0.999925497999794 13422.4584204016

62 9.71406393231318E-06 0.999935212063726 15434.9722728985

63 1.08212886005039E-05 0.999946033352327 18529.963285093

64 7.42420751202246E-06 0.999953457559839 21485.7664647741

65 8.1851440488331E-06 0.999961642703888 26070.6593362802

66 5.4913330684333E-06 0.999967134036956 30426.6148742469

67 6.00205486730809E-06 0.999973136091824 37224.6656531615

68 3.92557058846365E-06 0.999977061662412 43595.1383212284

69 4.26510791921652E-06 0.999981326770331 53552.6000453633

70 2.71842033462898E-06 0.999984045190666 62677.0260342233

71 2.9488195842364E-06 0.99998699401025 76887.6509388058

72 1.84159491604572E-06 0.999988835605166 89570.4617118085

73 2.00717881270556E-06 0.999990842783979 109203.495657722

74 1.24624227589492E-06 0.999992089026255 126406.689265949

75 1.37287111106041E-06 0.999993461897366 152949.57206274

76 8.6214931218563E-07 0.999994324046678 176181.857613473

77 9.5646317543291E-07 0.999995280509853 211887.294801416

78 6.1692584468004E-07 0.999995897435698 243749.988154011

79 6.8556788647498E-07 0.999996583003584 292654.68218478

80 4.5516863686756E-07 0.999997038172221 337629.354153593

81 5.0398897450652E-07 0.999997542161196 406861.507092043

82 3.3840507784668E-07 0.999997880566274 471824.142290239

83 3.7244496005772E-07 0.999998253011234 572413.526296464

84 2.4581612489222E-07 0.999998498827359 666145.899917703

85 2.6927196379775E-07 0.999998768099323 811753.754487664

86 1.7495810055047E-07 0.999998943057423 946125.193512013

87 1.9114497290999E-07 0.999999134202396 1155004.35156325

88 1.2250293111505E-07 0.999999256705327 1345361.45182769

89 1.3381714387489E-07 0.999999390522471 1640749.58104408

90 8.499680623592E-08 0.999999475519278 1906647.77000187

91 9.307751441182E-08 0.999999568596792 2318017.06870306

92 5.903763220078E-08 0.999999627634424 2685532.88661233

93 6.492749000821E-08 0.999999692561914 3252687.43833053

94 4.147165110268E-08 0.999999734033565 3759872.93616266

95 4.578647702527E-08 0.999999779820042 4541739.44984715

96 2.959960466574E-08 0.999999809419647 5247130.58808574

97 3.270704081475E-08 0.999999842126688 6334192.81611743

98 2.139282805491E-08 0.999999863519516 7327054.89966352

99 2.360027383067E-08 0.99999988711979 8858948.76905362

100 1.551956565477E-08 0.999999902639355 10271090.5779024

101 1.707166537407E-08 0.999999919711021 12455009.5097791

102 1.118180756664E-08 0.999999930892828 14470278.2084254

103 1.226777836731E-08 0.999999943160607 17593431.9824048

104 7.95295244473E-09 0.999999951113559 20455569.7179842

105 8.71404542047E-09 0.999999959827604 24892715.1847711

106 5.60660927147E-09 0.999999965434214 28930341.5015913

107 6.14400356617E-09 0.999999971578217 35184281.3793566

108 3.93655779041E-09 0.999999975514775 40840956.1965428

109 4.31969429127E-09 0.999999979834469 49589570.2119679

110 2.76759457198E-09 0.999999982602064 57478082.1543487

111 3.04296288819E-09 0.999999985645027 69662268.634202

112 1.95699208752E-09 0.999999987602019 80658293.721089

113 2.15538468698E-09 0.999999989757404 97631494.7177023

114 1.39429171123E-09 0.999999991151695 113015998.841094

115 1.53653941578E-09 0.999999992688235 136765877.784375

116 9.9914512108E-10 0.99999999368738 158412825.04409

117 1.1002284924E-09 0.999999994787608 191850507.366826

118 7.1709199153E-10 0.9999999955047 222454573.598716

119 7.885234784E-10 0.999999996293224 269776191.193109

120 5.1315516219E-10 0.999999996806379 313124179.261904

121 5.6352302502E-10 0.999999997369902 380213949.078925

122 3.6513042556E-10 0.999999997735032 441507386.433156

123 4.0069160606E-10 0.999999998135724 536401211.294286

124 2.5844390151E-10 0.999999998394168 622730053.616445

125 2.8364905263E-10 0.999999998677817 756324876.017021

126 1.8256568131E-10 0.999999998860382 877487393.739961

127 2.0051551448E-10 0.999999999060898 1064847035.18611

128 1.2908402761E-10 0.999999999189982 1234540555.99299

129 1.4191196933E-10 0.999999999331894 1496768443.35221

130 9.154410651E-11 0.999999999423438 1734419121.0406

131 1.0072018155E-10 0.999999999524158 2101539233.13154

132 6.515900105E-11 0.999999999589317 2434969808.81057

133 7.170967234E-11 0.999999999661027 2950086844.92162

134 4.650452123E-11 0.999999999707531 3419170411.04716

135 5.116302727E-11 0.999999999758694 4144123822.5676

136 3.32131941E-11 0.999999999791908 4805559352.03531

137 3.651472981E-11 0.999999999828422 5828265234.71141

138 2.368678808E-11 0.999999999852109 6761745339.05645

139 2.602407512E-11 0.999999999878133 8205684412.6793

140 1.684800007E-11 0.999999999894981 9522107201.67137

141 1.850398759E-11 0.999999999913485 11558716589.5944

142 1.195331629E-11 0.999999999925439 13411753085.5745

143 1.312899803E-11 0.999999999938568 16278051330.193

144 8.47049199E-12 0.999999999947038 18881473537.26

145 9.30712178E-12 0.999999999956345 22906958288.5928

146 6.00558608E-12 0.999999999962351 26560977295.9565

147 6.60192116E-12 0.999999999968953 32208944980.1036

148 4.26463412E-12 0.999999999973217 37337552924.0581

149 4.68983714E-12 0.999999999977907 45263445085.2584

150 3.03384962E-12 0.999999999980941 52468408709.4408

151 3.33673655E-12 0.999999999984278 63603875709.612

152 2.16105481E-12 0.999999999986439 73739443259.7974

153 2.37643298E-12 0.999999999988815 89406805911.4289

154 1.53980319E-12 0.999999999990355 103679991421.479

155 1.69270348E-12 0.999999999992048 125749696413.986

156 1.0963684E-12 0.999999999993144 145858488733.195

157 1.20483738E-12 0.999999999994349 176955251463.448

158 7.7963436E-13 0.999999999995128 205273576306.228

159 8.5661694E-13 0.999999999995985 249072235565.108

160 5.5373002E-13 0.999999999996539 288923793255.525

161 6.0842437E-13 0.999999999997147 350542878176.338

162 3.930462E-13 0.99999999999754 406553791683.186

163 4.3195046E-13 0.999999999997972 493166844871.933

164 2.7905017E-13 0.999999999998251 571849359071.868

165 3.0674451E-13 0.999999999998558 693501636490.683

166 1.9827553E-13 0.999999999998756 804070635131.315

167 2.1799257E-13 0.999999999998974 975016156607.598

168 1.4100999E-13 0.999999999999115 1130421593215.49

169 1.5504057E-13 0.99999999999927 1370541578627.66

170 1.0034621E-13 0.999999999999371 1589131837463.13

171 1.1032189E-13 0.999999999999481 1927085848254.38

172 7.141763E-14 0.999999999999552 2234482573738.77

173 7.85047E-14 0.999999999999631 2709747068213.3

174 5.081036E-14 0.999999999999682 3142777130056.17

175 5.584336E-14 0.999999999999738 3811764390495.55

176 3.612632E-14 0.999999999999774 4419626719696.27

177 3.970116E-14 0.999999999999814 5361428127822.02

178 2.567058E-14 0.999999999999839 6216148553996.54

179 2.821102E-14 0.999999999999867 7537405234092.88

180 1.823534E-14 0.999999999999885 8736371731077.59

181 2.004168E-14 0.999999999999906 10596705005577.6

182 1.295462E-14 0.999999999999919 12288129951897.7

183 1.42394E-14 0.999999999999933 14887932652464.4

184 9.20639707105513E-15 0.999999999999942 17255170986093.9

185 1.012019E-14 0.999999999999952 20898374140930.4

186 6.54535576278727E-15 0.999999999999959 24212901222422

187 7.19504812515522E-15 0.999999999999966 29339411253228

188 4.65460590914123E-15 0.999999999999971 33989431149966

189 5.11626962030467E-15 0.999999999999976 41128763720278.5

190 3.30992993545501E-15 0.999999999999979 47657138913973.5

191 3.63779601445147E-15 0.999999999999983 57738456761160.2

192 2.35301386063407E-15 0.999999999999985 66719994479562.9

193 2.58575823862323E-15 0.999999999999988 80421421917330.3

194 1.67196653051699E-15 0.999999999999989 92857724275680.3

195 1.83714396646461E-15 0.999999999999991 112589990684262

196 1.18745160409666E-15 0.999999999999992 130539119633927

197 1.3046575357082E-15 0.999999999999994 158021039556860

198 8.42993824544766E-16 0.999999999999995 183820392953898

199 9.26139465134362E-16 0.999999999999995 219687786701000

200 5.98273386137908E-16 0.999999999999996 1671476657946290

Finally, here is how these probabilities relate to various events from games, sports, ways of dying, etc. in the USA. Along with the lowest roll needed to represent them happening.

Description Roll
Scoring above 150 in Yahtzee  3

Congressperson being reelected 4

Scoring above 200 in Yahtzee 7

Random person having a 100 IQ (15 S.D.) 11

High card in poker 11

One pair in poker 12

Scoring above 250 in Yahtzee 12

Random person having a 116 IQ (15 S.D.) 17

Scoring above 300 in Yahtzee 17

Scoring above 350 in Yahtzee 19

Random person having a 124 IQ (15 S.D.) 21

Two pairs in poker 23

Natural 21 Blackjack 23

Having an item lost in the mall returned 25

Winning the Powerball 25

Scoring above 400 in Yahtzee 25

Random person qualifying for MENSA 29

Three of a kind in poker 29

Buster Douglas knocking out Mike Tyson in 1990 29

Scoring above 450 in Yahtzee 32

Death by motor vehicle 33

Being ambidextrous 33

Boston Red Sox  winning the 2004 world series 34

Scoring above 500 in Yahtzee 35

Death by murder 36

Death by fire 37

Rolling 3 sixes with 3D6. 37

Straight in poker 37

Random person having a 140 IQ (S.D. 15) 37

Getting a car stolen in lifetime 39

Flush in poker 42

Full house in poker 44

Random person having a 150 IQ (S.D. 15) 51

Death by firearms accident 52

Random person getting a perfect SAT 53

Four of a kind in poker 54

Navigating an asteroid field (3,720 to 1) 54

Leicester City Foxes winning 15-16 Premier League 55

Being killed by falling furniture 56

Death by drowning 59

Clover being a four leaf clover 60

Death by flood 66

Death by tornado 70

Killed by bees 70

Straight flush in poker 71

Death by earthquake 75

Death by lightning 75

Royal flush in poker 84

Death by botulism 93

Death by shark attack 99

Winning Mega Millions 120

Creating a perfect March Madness bracket 130


Sharing

By Sébastien Andrivet.

Helper(s): Jasper at http://catlikecoding.com/anydice/, who helped me spot a really stupid mistake of mine. It’s a cool site, you should check it out. Also, Eric Langendorff, Vincent Primault, Adrian Patten.