# DC Heroes RPG – A short article on probabilities

## 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*.

## 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#), or to 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 very 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.

## 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

The “explodes on a double” mechanic (that is, reroll and add the new roll to the original roll if you get a double) has 2 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:

The calculations in the original version of the article assumed one re-roll after a double, to keep things simple. However, one of the Teeming Million of writeups.org readers kindly simulated a hundred million MEGS dice rolls to double-check and to simulate a process where one keeps rolling as long as doubles come up.

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.

Thus, when assessing values for, say, combat, getting an idea of the AV vs. OV probabilities from the tables above is important, but one should be careful to remember that:

- People are spending HPs (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;

}

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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