Odds

[ArcticShadow]

So I got bored and wrote some code simulating 1,000,000 dice rolls to get a win%. Here are my findings (since I used simulations and not exact math-- the numbers may be slightly off the true %.) I didn't bother with the shield boost numbers.

All numbers rounded to the nearest .5 (with one exception in 3v7)


http://imgur.com/eND8O
* means that if you win you are ridiculously lucky
! means that if you lose you are ridiculously unlucky.

Put the numbers in excel. Didn't feel like formatting it into here so hopefully the image suffices.

[Ratburntro44]

So all the numbers are the chances of offense winning, right?

[Lopdo]

yes

[Darkas]

Here are the exact numbers rounded to the nearest .1%, without any kind of format because I'm lazy.

There is one more column because #defenders can be 1.

Code: Select all

83.8   44.4   15.2   3.6    0.6    0.1    0      0
97.3   77.9   45.4   19.2   6.1    1.5    0.3    0
99.7   93.9   74.3   46     22     8.3    2.5    0.6
100    98.8   90.9   71.8   46.4   24.2   10.4   3.7
100    99.8   97.5   88.4   70     46.7   26     12.2
100    100    99.5   96.2   86.2   68.5   46.9   27.4
100    100    99.9   99     94.8   84.4   67.3   47.1


[ArcticShadow]

Aye, realized after all was said and done D can have 1 guy. Didn't feel like going back and adding it in. Also, out of curiosity-- whats the math formula to get those? Thought about it it couldn't devise a way, which is why I went the route of a simulation.

[techgump]

So how about posting one for defense (shield boost) too?

[Darkas]

There is no math formula actually
I mean, I didn't use any.

It would be an awful one...

I bruteforced all the possible permutation of the 1 to 8 dices in some tables and used them to compute all possible combinations in a given battle for the attacker to win and divided by the total of possibilites to get the probability

[techgump]

So, doesn't it matter how many sides the dice have? As I posted in the totally defunct thread on the same topic, it seems to me that the number or sides does factor into odds... as more die sides means the odds of the defender winning becomes less, while the attacker more. I am not saying the attacker gains the overall advantage due to this... they cant, because on tied rolls the defender wins. But the advantage spread is less. The change in spread is slight if just adding one more side, but is more dramatic when comparing say a 6 sided die to a 30 sided die.

[Darkas]

It's 6 sides dice (and you're right, it does matter, the more sides, the more close to 50/50 the odds are, because equality is less likely)

Attack boost:

Code: Select all

97.4    69.1    28.1    7.0    1.2     0.2    0.0    0.0
99.9    95.1    70.1    34.7   11.8    3.0    0.6    0.1
100.0   99.6    93.4    70.8   39.2    16.0   5.0    1.3
100.0   100.0   99.2    92.1   71.2    42.6   19.7   7.2
100.0   100.0   99.9    98.7   91.0    71.6   45.2   22.8
100.0   100.0   100.0   99.9   98.1    90.1   71.8   47.3
100.0   100.0   100.0   100.0  99.7    97.6   89.3   72.0


Defense boost:

Code: Select all

70.2    19.7    2.3    0.1    0.0    0.0    0.0    0.0
94.7    60.6    20.6   3.7    0.4    0.0    0.0    0.0
99.5    88.2    55.2   21.1   4.9    0.7    0.1    0.0
100.0   97.6    82.7   51.6   21.5   5.9    1.1    0.1
100.0   99.6    95.1   78.1   48.9   21.8   6.8    1.5
100.0   100.0   98.9   92.5   74.4   46.9   22.0   7.5
100.0   100.0   99.8   97.9   89.8   71.2   45.4   22.2


Column: #defenders
Row: #attacker - 1

[techgump]

Here is what I came up with to deduce the affect of the number of die sides.

If you had a 2 sided die with "1"s and "2"s:
Possible roll combinations is 4. (1v1, 1v2, 2v1, 2v2)
Total possible combinations for an attacker win is 1.
Total possible combinations for a defender win is 3.

If you had a 3 sided die with "1"s, "2"s, and "3"s:
Possible roll combinations is 9. (1v1, 1v2, 1v3, 2v1, 2v2, 2v3, 3v1, 3v2, 3v3)
Total possible combinations for an attacker win is 3.
Total possible combinations for a defender win is 6.

If you had a 4 sided die with "1"s, "2"s, "3"s, and "4"s:
Possible roll combinations is 16. (1v1, 1v2, 1v3, 1v4, 2v1, 2v2, 2v3, 2v4, 3v1, 3v2, 3v3, 3v4, 4v1, 4v2, 4v3, 4v4)
Total possible combinations for an attacker win is 6.
Total possible combinations for a defender win is 10.

If you had a 5 sided die with "1"s, "2"s, "3"s, "4"s, and "5"s:
Possible roll combinations is 25. (1v1, 1v2, 1v3, 1v4, 1v5, 2v1, 2v2, 2v3, 2v4, 2v5, 3v1, 3v2, 3v3, 3v4, 3v5, 4v1, 4v2, 4v3, 4v4, 4v5, 5v1, 5v2, 5v3, 5v4, 5v5)
Total possible combinations for an attacker win is 10.
Total possible combinations for a defender win is 15.

I'll stop there, as this now clearly illustrates the repetition by adding another side to each die, and the output ramifications.

Die Side | Total Roll Comb | Attacker Win Combinations | Defender Win Combinations
2 | 4 | 1 | 3
3 | 9 | 3 | 6
4 | 16 | 6 | 10
5 | 25 | 10 | 15

Of course there are equations for this that can be deduced by looking at it chart:
For each side die with X sides:
Total Roll Comb: x^2
Attacker Win Comb: x(x-1)/2
Defender Win Comb: x^2 - x(x-1)/2

Which means with a 6 sided die, and assuming equals on attack/defence (4x4 or 8x8):
Attacker:
15/36 = 41.7%
Defender:
21/36 = 58.3%


Likewise, a 30 sided die:
Attacker:
435/900 = 48.3%
Defender:
465/900 = 51.7%

Almost a 9% gain for attacker (and loss for defender) when using die with 24 more sides.