Massive darkness vanilla dice distributions

Picking as source of the dice (since I don't have the game but I played it) this post:
https://www.reddit.com/r/MassiveDarkness/comments/6d1fqu/massive_darkness_dice_distributions/

Impressive how difficult is to find a source online with the dice distributions, but the post above should have it right. I report it here for simplicity (and moreover if the above source gets lost)

Attack dice

Red (strong)

  • Blank
  • 1 Sword
  • 1 Sword
  • 2 Sword / 1 Bam
  • 2 Sword / 1 Bam
  • 3 Sword / 1 Diamond

Yellow (weak)

  • Blank
  • 1 Sword
  • 1 Sword
  • 1 Sword
  • 1 Sword
  • 2 Sword / 1 Bam

Defense dice

Green (strong)

  • Blank
  • Blank
  • 1 Shield
  • 2 Shield / 1 Bam
  • 2 Shield / 1 Bam
  • 3 Shield / 1 Diamond

Blue (weak)

  • Blank
  • Blank
  • 1 Shield
  • 1 Shield
  • 1 Shield
  • 2 Shield / 1 Bam

Observations

One can observe that the defensive dice areweaker than their attacking counterpart.

On boardgamegeek (and other internet places) a lot of people wrote their experience and observations. I read a few quite quickly but I agree with the following idea. The human controlled characters can attack multiple times, while they get only one counterattack if any (if the opposition is not stunned). Therefore having attacking dice more powerful than defensive one, the game tends - aside from unlucky or reckless action - to be clearly feasible without much difficulty. The only difficulty is it that may take a while to bring down all the enemies.

The expected value is the first "important aggregated value" as it represents the average damage dealt with a certain dice combination over time. One can clearly see that attack is superior to defense with dice alone!

Distributions of results.

Percentages rounded to two decimal digits.
Remember that one may use normally - withot abilities - 3 dice of one color.
Results ordered by increasing strength.

Some results unformatted due to lack of time. My approach is, though, better to have the data, not so neatly formatted, than having no date at all. One can always improve the formatting later, while without content is difficult to format anything.

Attack

1 Yellow

Yellow frequency (perc)
Blank 1/6 (16.67%)
1 sword 4/6 (66.67 %)
2 swords , 1 bam 1/6 (16.67%)
1 sword , 0.17 bams expected value

2 Yellow

2 Yellow frequency (perc)
Blank 1/36 (2.78%)
1 sword 8/36 (22.22 %)
2 swords 16/36 (44.44 %)
2 swords , 1 bam 2/36 (5.56%)
3 swords , 1 bam 8/36 (22.22%)
4 swords , 2 bams 1/36 (2,78%)
2 swords , 0.33 bams expected value

The ones with 1/36 have to happen to both dice.
Then there are the ones that have one dice blank, and the other not blank.
And then similarly for the rest.

3 Yellow

I can see the 2 yellow as a dice and combine it with the other yellow.
Ok after getting almost right with pen, paper and scientific calculator I gave it a try with the 50g, nice experience. See https://app.assembla.com/spaces/various-works-only-code/git-2/source/master/rpl_hp48-50/programs/general/urpl.diceMassiveDark and http://www.hpmuseum.org/forum/thread-8555-post-101689.html#pid101689

3 Yellow frequency (perc)
Blank 1/216 (0.46%)
1 sword 12/216 (5.56 %)
2 swords 48/216 (22.22 %)
3 swords 64/216 (29.63 %)
2 swords , 1 bam 3/216 (1.38%)
3 swords , 1 bam 24/216 (11.11%)
4 swords , 1 bam 48/216 (22.22%)
4 swords , 2 bams 3/216 (1.38%)
5 swords , 2 bams 12/216 (5.56%)
6 swords , 3 bams 1/216 (0.46%)
3 swords , 0.5 bams expected value

1 Red

Red frequency (perc)
Blank 1/6 (16.67%)
1 sword 2/6 (33.33 %)
2 swords , 1 bam 2/6 (33.33%)
3 swords , 1 diamond 1/6 (16.67%)
1.5 swords , 0.33 bams, 0.17 diamonds expected value

2 Red

2 Red frequency (perc)
Blank 1/36 (2.78%)
1 sword 4/36 (11.11%)
2 swords 4/36 (11.11%)
2 swords , 1 bam 4/36 (11.11%)
3 swords , 1 bam 8/36 (22.22%)
3 swords , 1 diamond 2/36 (5.55%)
4 swords , 1 diamond 4/36 (11.11%)
4 swords , 2 bams 4/36 (11.11%)
5 swords , 1 bam, 1 diamond 4/36 (11.11%)
6 swords , 2 diamonds 1/36 (2.78%)
3 swords , 0.67 bams, 0.33 diamonds expected value

3 Red

3 Red frequency (perc)
Blank 1/216 (0.46%)
1 sword 6/216 (2.78%)
2 swords 12/216 (5.55%)
3 swords 8/216 (3.70%)
2 swords , 1 bam 6/216 (2.78%)
3 swords , 1 bam 24/216 (11.11%)
3 swords , 1 diamond 3/216 (1.39%)
4 swords , 1 bam 24/216 (11.11%)
4 swords , 2 bams 12/216 (5.55%)
4 swords , 1 diamond 12/216 (5.55%)
5 swords , 2 bams 24/216 (11.11%)
5 swords , 1 diamond 12/216 (5.55%)
5 swords , 1 bam, 1 diamond 12/216 (5.55%)
6 swords , 3 bams 8/216 (3.70%)
6 swords , 1 bam, 1 diamond 24/216 (11.11%)
6 swords , 2 diamonds 3/216 (1.39%)
7 swords , 2 bams 6/216 (2.78%)
7 swords , 2 bams, 1 diamond 12/216 (5.55%)
8 swords , 1 bam, 2 diamonds 6/216 (2.78%)
9 swords , 3 diamonds 1/216 (0.46%)
4.5 swords , 1 bams, 0.5 diamonds expected value

1 Yellow 1 Red

1 Yellow 1 Red frequency (perc)
blank 1/36 (-%)
1 sword 6/36 (-%)
2 swords 8/36 (-%)
2 swords, 1 bam 3/36 (-%)
3 swords, 1 bam 10/36 (-%)
3 swords, 1 diamond 1/36 (-%)
4 swords, 2 bams 2/36 (-%)
4 swords, 1 diamond 4/36 (-%)
5 swords, 1 bam, 1 diamond 1/36 (-%)
2.5 swords , 0.5 bams, 0.17 diamonds expected value

2 Yellow 1 Red

2 Yellow 1 Red frequency (perc)
blank 1/216 (-%)
1 sword 10/216 (-%)
2 swords 32/216 (-%)
2 swords, 1 bam 4/216 (-%)
3 swords 32/216 (-%)
3 swords, 1 bam 28/216 (-%)
3 swords, 1 diamond 1/216 (-%)
4 swords, 1 bam 48/216 (-%)
4 swords, 2 bams 5/216 (-%)
4 swords, 1 diamond 8/216 (-%)
5 swords, 2 bams 18/216 (-%)
5 swords, 1 diamond 16/216 (-%)
5 swords, 1 bam, 1 diamond 2/216 (-%)
6 swords, 3 bams 2/216 (-%)
6 swords, 1 bam, 1 diamond 8/216 (-%)
7 swords, 2 bams, 1 diamond 1/216 (-%)
3.5 swords , 0.67 bams, 0.17 diamonds expected value

3 Yellow 1 Red

3 Yellow 1 Red frequency (perc)
4.5 swords , 0.83 bams, 0.17 diamonds expected value

:frequency: {swords bams diamonds}

:3Y1R:
:1.: { 0. 0. 0. }
:14.: { 1. 0. 0. }
:72.: { 2. 0. 0. }
:160.: { 3. 0. 0. }
:128.: { 4. 0. 0. }
:5.: { 2. 1. 0. }
:54.: { 3. 1. 0. }
:1.: { 3. 0. 1. }
:192.: { 4. 1. 0. }
:9.: { 4. 2. 0. }
:12.: { 4. 0. 1. }
:224.: { 5. 1. 0. }
:66.: { 5. 2. 0. }
:48.: { 5. 0. 1. }
:3.: { 5. 1. 1. }
:120.: { 6. 2. 0. }
:7.: { 6. 3. 0. }
:64.: { 6. 0. 1. }
:24.: { 6. 1. 1. }
:26.: { 7. 3. 0. }
:48.: { 7. 1. 1. }
:3.: { 7. 2. 1. }
:2.: { 8. 4. 0. }
:12.: { 8. 2. 1. }
:1.: { 9. 3. 1. }
exp_val {4.5 0.83 0.17}

1 Yellow 2 Red

1 Yellow 2 Red frequency (perc)
blank 1/216 (-%)
1 sword 8/216 (-%)
2 swords 20/216 (-%)
3 swords 16/216 (-%)
2 swords, 1 bam 5/216 (-%)
3 swords, 1 bam 28/216 (-%)
3 swords, 1 diamond 2/216 (-%)
4 swords, 1 bam 36/216 (-%)
4 swords, 2 bams 8/216 (-%)
4 swords, 1 diamond 12/216 (-%)
5 swords, 2 bams 24/216 (-%)
5 swords, 1 diamond 16/216 (-%)
5 swords, 1 bam, 1 diamond 6/216 (-%)
6 swords, 3 bams 4/216 (-%)
6 swords, 1 bam, 1 diamond 20/216 (-%)
6 swords, 2 diamonds 1/216 (-%)
7 swords, 2 bams, 1 diamond 4/216 (-%)
7 swords, 2 diamonds 4/216 (-%)
8 swords, 1 bam, 2 diamonds 1/216 (-%)
4 swords , 0.83 bams, 0.33 diamonds expected value

2 Yellow 2 Red

Note that with independent events like dice, we can make the sum of the expected values.
Therefore:
2 Yellow 2 Red frequency (perc)
5 swords , 1 bam, 0.33 diamonds expected value

3 Yellow 2 Red

3 Yellow 2 Red frequency (perc)
6 swords , 1.17 bam, 0.33 diamonds expected value

1 Yellow 3 Red

1 Yellow 3 Red frequency (perc)
5.5 swords , 1.17 bam, 0.5 diamonds expected value

2 Yellow 3 Red

2 Yellow 3 Red frequency (perc)
6.5 swords , 1.33 bam, 0.5 diamonds expected value

3 Yellow 3 Red

2 Yellow 3 Red frequency (perc)
7.5 swords , 1.5 bam, 0.5 diamonds expected value

Defense

1 Blue

1 Blue frequency (perc)
Blank 2/6 (-%)
1 shield 3/6 (-%)
2 shields , 1 bam 1/6 (-%)
0.83 shields , 0.17 bams expected value

2 Blue

2 Blue frequency (perc)
1.67 shields , 0.33 bams expected value

:2B:
:4.: { 0. 0. 0. }
:12.: { 1. 0. 0. }
:9.: { 2. 0. 0. }
:4.: { 2. 1. 0. }
:6.: { 3. 1. 0. }
:1.: { 4. 2. 0. }
exp_val { 1.67 .33 0. }

3 Blue

3 Blue frequency (perc)
2.5 shields , 0.5 bams expected value

:3B:
:8.: { 0. 0. 0. }
:36.: { 1. 0. 0. }
:54.: { 2. 0. 0. }
:27.: { 3. 0. 0. }
:12.: { 2. 1. 0. }
:36.: { 3. 1. 0. }
:27.: { 4. 1. 0. }
:6.: { 4. 2. 0. }
:9.: { 5. 2. 0. }
:1.: { 6. 3. 0. }
exp_val { 2.5 .5 0. }

1 Green

Green frequency (perc)
Blank 2/6 (-%)
1 shield 1/6 (-%)
2 shields , 1 bam 2/6 (-%)
3 shields , 1 diamond 1/6 (-%)
1.33 shields , 0.33 bams, 0.17 diamonds expected value

2 Green

2 Green frequency (perc)
2.67 shields , 0.67 bams, .33 diamonds expected value

:2G:
:4.: { 0. 0. 0. }
:4.: { 1. 0. 0. }
:1.: { 2. 0. 0. }
:8.: { 2. 1. 0. }
:4.: { 3. 1. 0. }
:4.: { 3. 0. 1. }
:4.: { 4. 2. 0. }
:2.: { 4. 0. 1. }
:4.: { 5. 1. 1. }
:1.: { 6. 0. 2. }
exp_val { 2.67 .67 .33 }

3 Green

3 Green frequency (perc)
4 shields , 1 bams, .5 diamonds expected value

:3G:
:8.: { 0. 0. 0. }
:12.: { 1. 0. 0. }
:6.: { 2. 0. 0. }
:1.: { 3. 0. 0. }
:24.: { 2. 1. 0. }
:24.: { 3. 1. 0. }
:12.: { 3. 0. 1. }
:6.: { 4. 1. 0. }
:24.: { 4. 2. 0. }
:12.: { 4. 0. 1. }
:24.: { 5. 1. 1. }
:8.: { 6. 3. 0. }
:12.: { 6. 1. 1. }
:6.: { 6. 0. 2. }
:12.: { 5. 2. 0. }
:3.: { 5. 0. 1. }
:12.: { 7. 2. 1. }
:3.: { 7. 0. 2. }
:6.: { 8. 1. 2. }
:1.: { 9. 0. 3. }
exp_value { 4. 1. .5 }

1 Blue 1 Green

1 Blue 1 Green frequency (perc)
2.16 shields, 0.5 bam, 0.17 diamonds expected value

2 Blue 1 Green

2 Blue 1 Green frequency (perc)
2.16 shields, 0.5 bam, 0.17 diamonds expected value

3 Blue 1 Green

3 Blue 1 Green frequency (perc)
3 shields, 0.67 bam, 0.17 diamonds expected value

1 Blue 2 Green

1 Blue 2 Green frequency (perc)
3.5 shields, 0.83 bam, 0.33 diamonds expected value

2 Blue 2 Green

2 Blue 2 Green frequency (perc)
4.32 shields, 1 bam, 0.33 diamonds expected value

3 Blue 2 Green

3 Blue 2 Green frequency (perc)
5.15 shields, 1.17 bam, 0.33 diamonds expected value

1 Blue 3 Green

1 Blue 3 Green frequency (perc)
4.82 shields, 1.17 bam, 0.5 diamonds expected value

2 Blue 3 Green

2 Blue 3 Green frequency (perc)
5.65 shields, 1.33 bam, 0.5 diamonds expected value

3 Blue 3 Green

2 Blue 3 Green frequency (perc)
6.5 shields, 1.5 bam, 0.5 diamonds expected value

Theory to derive the frequency

To be organizing but better some links than nothing.

- binomial coefficent and combinations to count how many times a certain result can happen with the dice. https://en.wikipedia.org/wiki/Combination . Although another way is to enumerate all the possible combinations with repetitions and then count the repetitions.
- https://en.wikipedia.org/wiki/Law_of_total_probability to sum different probabilities in different cases.
- expected value: https://en.wikipedia.org/wiki/Expected_value

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