# Thread: Comprehensive probabilities index

1. Senior Member
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## Will someone please double-check my work on Rukar? Was not working with known formulas, just logic.

R. RUKAR
1. 0 damage: 3.7%

2. 1 damage: 5.6%
- Probability of one hit is 22.2%
- Probability of hit being six is 25%
- Product of both probabilities: 5.6%

3. 2 damage: 30.6%
One six-hit and two misses: 5.6%
- Probability of rolling 1 hit out of 3 dice: 22.2%
- Probability 1 hit will be a 6: 25%
- Product of both is 5.6%

Two non-six hits and one miss:
- Probability of two hits is 44.4%
- Probability both hits will not be a six: 56.3%
- Product of both is: 25%

Sum of above two products is 30.6%

4. 3 damage: 29.2%

Probability of two hits is 2: 44.4%
Probability one of those two will be a 6: 37.5
Product of those two numbers is 16.7%

Probability of 3 hits is 29.6%
Probability all 3 are not sixes is 42.2
Product of those two numbers is 12.5%

Sum of those two products is 29.2%

5. 4 damage: 15.3%
Probability of two hits is 2: 44.4%
Probability both dice will be sixes: 6.3%
Product of those two probabilities is: 2.8%

Probability three hits is: 29.6%
Probability one out of three is a six is: 42.2%
Product of those two probabilities: 12.5

Sum of two products is 15.3%

6. 5 damage: 12.4%
Probability of three hits is 29.6%
Probability one of three hits will be a six: 42%
Product of those two probabilities is: 12.4%

6. 6 damage 0.5%

My total probabilities sum comes up to 102.7: not sure if that's just because I rounded all tenths up.
Last edited by commandercool; 01-02-2017 at 03:00 PM.  Reply With Quote

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## Rukar's probabilities are:
0 damage - 3.70%
1 damage - 11.11%
2 damage - 22.22%
3 damage - 25.93%
4 damage - 22.22%
5 damage - 11.11%
6 damage - 3.70%

I think part of the problem in your math might have been that he places an extra wound when he rolls 5+, not just on sixes. The result table is bell shaped, because there are equal chances of rolling each of the 3 outcomes for each die.  Reply With Quote

3. Senior Member
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## here's a more in depth analysis:

Rukar gets 3 dice, so there are 216 different combinations possible (6x6x6=216).

The standard hit probability for 3 dice is:

1. 8/216 times there is no hit (3.7%)
2. 48/216 times there is one hit (22.2%)
3. 96 /216 there is two hits (44.4%)
4. 64/216 there is three hits (29.6%)

However, of those combinations that we have listed, you know that a portion of them are going to have extra hits. There are exactly 4 numbers that normally equal one hit: 3,4,5 and 6. In Rukar's case, you know that 50.0% of the time you roll a hit, it contains a 5 or 6. So # 2 of the above list is modified like this:

2. 48/216= one hit
2. a. 24/216 = 3 or 4
2. b. 24/216 = 5 or 6

You know that 96 times, you will roll 2 hits. You know from the first example that there is a 50.0% chance that the first dice will contain a 5 or 6. You know that there is a 50.0% chance the second dice will also contain a 5 or 6.
So:

3. 96/216 there is two hits.
3. a. 48/216 = first dice is a 5 or 6.
3. b. 48/216 = first dice is a 2 or 3.
3.i. 48/216 = second dice is a 5 or 6.
3. ii. 48/216 = second dice is a 2 or 3.

Notice here that these fractions added up do not equal 216. That is because you have broken what should have been a single combination into separate components, namely first roll and second roll. In other words, (i) and (ii) need to be inserted back into (a) and (b), and more importantly (i) and (ii) are evenly distributed between (a) and (b):

3. 96/216 there is two hits.
3. a. 48/216 = first dice is a 5 or 6.
3. a. i. 24/216 = second dice is a 5 or 6.
3. a. ii. 24/216 = second dice is a 2 or 3.
3.b. 48/216 = first dice is a 2 or 3.
3. b.i. 24/216 = second dice is a 5 or 6.
3. b.ii. 24/216 = second dice is a 2 or 3.

With three dice, the exercise is the same, you just need to add one more step to include one more roll.

4. 64/216 there is three hits (29.6%)
4.a. 32/216 = first dice is a 5 or 6.
4. a.i. 16/216 = second dice is a 5 or 6.
4.a.i.x: 8/216 = third dice is a 5 or 6.
4. a.i. y. 8/216 = third dice is a 2 or 3.
4.a.ii. 16/216 = second dice is a 2 or 3.
4.a.ii.x. 8/216 = third dice is a 5 or 6.
4.a.ii.y. 8/216 = third dice is a 2 or 3.
4. b. 32/216 times the first dice rolled is a 2 or 3.
4.b.i. 16/216 = second dice is a 5 or 6.
4.b.i.x. 8/216 = third dice is a 5 or 6.
4.b.i.y. 8/216 = third dice is a 2 or 3.
4.b.ii. 16/216 = second dice is a 2 or 3.
4.b.ii.x. 8/216 = third dice is a 5 or 6.
4.b.ii.y. 8/216= third dice is a 2 or 3.

Now you have to count how many wounds each combination results in (assuming no modifiers), and add them together.

1. No wounds = 8/216 (#1.)
2. 1 wound only = 24/216 (#2a)
3. 2 wounds = 24/216 (#2a) + 24/216 (#3bii)
4. 3 wounds = 24/216(3aii) + 24/216 (3bi) + 8/216 (4biiy)
5. 4 wounds = 24/216 (3ai) + 8/216 (4aiiy) + 8/216 (4biy) + 8/216 (4biix)
6. 5 wounds = 8/216 (4aiy) + 8/216(4aiix) + 8/216 (4bix)
7. 6 wounds = 8/216(4aix)

So your probability for each hit is:

1. 0 = 8/216 = 3.7%
2. 1 = 24/216 = 11.1%
3. 2 = 48/216 = 22.2%
4. 3 = 56/216 = 25.9%
5. 4 = 48.216 = 22.2%
6. 5 = 24/216 = 11.1%
7. 6 = 8/216 = 3.7%

Compared to other characters with 3 dice that have an average attack damage of 2, rukar’s average damage is 3, putting him in exactly between 4 and 5 regular dice.

In looking over your work, jtberman is correct that you seem to have omitted 5s form the calculation.
Your second point multiplied 22.2% by 25% when it should have multiplied it by 75% to get one hit. And your 4th 5th and 6th points need to have the first roll, second roll, third roll analysis to get the right numbers.
Last edited by Rdebruys83; 01-02-2017 at 05:26 PM.  Reply With Quote

4. ## You could have made this problem a lot smaller by noting that each die has only 3 possibilities with equal chance: (1-2): 0 wounds, (3-4) 1 wound, (5-6) 2 wounds.
This means there are 3*3*3=27 possibilities.

Rukar (3AV Power Surge)
 Wounds Chance List of Possible Outcomes: 0=1-2; 1=3-4; 2=5-6; 0 1/27 (000) 1 3/27 (100), (010), (001) 2 6/27 (200), (020), (002), (110), (101), (011) 3 7/27 (210), (201), (120), (021), (102), (012), (111) 4 6/27 (220), (202), (022), (211), (121), (112) 5 3/27 (221), (212), (122) 6 1/27 (222)

Same answer, but it takes a lot fewer steps.
In addition, the width of the third column functions as a horizontal bar graph of the outcomes.
Last edited by vintageGamebro; 01-03-2017 at 11:54 AM.  Reply With Quote

5. Senior Member
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## Thanks for the help guys! @jy and @ Rdebruy, you were right, I calculated Rukar as if triggering on rolls of 6.

I'll be slowly plugging away at this project over the next couple months. Hope it will be useful.  Reply With Quote

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## I think your stats for TO Berserker with Raldag are wrong. They should be the same as a normal 3 dice unit:

Exactly 0 Hits = 3.70%
Exactly 1 Hits = 22.22% At least 1 Hits = 96.30%
Exactly 2 Hits = 44.44% At least 2 Hits = 74.07%
Exactly 3 Hits = 29.63% At least 3 Hits = 29.63%  Reply With Quote

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## TO Blagog with Rune of Power (ndice:5, hitmap:[0,0,0,1,2,2])

Exactly 0 Hits = 3.13%
Exactly 1 Hits = 5.21% At least 1 Hits = 96.87%
Exactly 2 Hits = 13.89% At least 2 Hits = 91.67%
Exactly 3 Hits = 15.05% At least 3 Hits = 77.78%
Exactly 4 Hits = 21.03% At least 4 Hits = 62.73%
Exactly 5 Hits = 15.44% At least 5 Hits = 41.71%
Exactly 6 Hits = 14.02% At least 6 Hits = 26.26%
Exactly 7 Hits = 6.69% At least 7 Hits = 12.24%
Exactly 8 Hits = 4.12% At least 8 Hits = 5.56%
Exactly 9 Hits = 1.03% At least 9 Hits = 1.44%
Exactly 10 Hits = 0.41% At least 10 Hits = 0.41%  Reply With Quote

8. Senior Member
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## FL Abomination

Exactly 0 Hits = 8.32%
Exactly 1 Hits = 24.83% At least 1 Hits = 91.68%
Exactly 2 Hits = 23.87% At least 2 Hits = 66.85%
Exactly 3 Hits = 20.67% At least 3 Hits = 42.98%
Exactly 4 Hits = 14.27% At least 4 Hits = 22.31%
Exactly 5 Hits = 6.58% At least 5 Hits = 8.05%
Exactly 6 Hits = 1.46% At least 6 Hits = 1.46%
Exactly 7 Hits = 0.00% At least 7 Hits = 0.00%  Reply With Quote

9. Senior Member
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## CL Vlox copying Slasher (ndice:2, hitmap:[0,0,1,1,1,3])

Exactly 0 Hits = 11.11%
Exactly 1 Hits = 33.33% At least 1 Hits = 88.89%
Exactly 2 Hits = 25.00% At least 2 Hits = 55.56%
Exactly 3 Hits = 11.11% At least 3 Hits = 30.56%
Exactly 4 Hits = 16.67% At least 4 Hits = 19.44%
Exactly 5 Hits = 0.00% At least 5 Hits = 2.78%
Exactly 6 Hits = 2.78% At least 6 Hits = 2.78%  Reply With Quote

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## TO Rukar with Rune of Heroism (ndice:4 hitmap:0,0,1,1,2,2)

Exactly 0 Hits = 1.23%
Exactly 1 Hits = 4.94% At least 1 Hits = 98.77%
Exactly 2 Hits = 12.35% At least 2 Hits = 93.83%
Exactly 3 Hits = 19.75% At least 3 Hits = 81.48%
Exactly 4 Hits = 23.46% At least 4 Hits = 61.73%
Exactly 5 Hits = 19.75% At least 5 Hits = 38.27%
Exactly 6 Hits = 12.35% At least 6 Hits = 18.52%
Exactly 7 Hits = 4.94% At least 7 Hits = 6.17%
Exactly 8 Hits = 1.23% At least 8 Hits = 1.23%  Reply With Quote

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