Advanced Probability Help

[giants756]

I need help creating a formula determining if the "ATTACKER" lands a Hit on the "DEFENDER". There are many factors that go into this and I can't seem to figure it out... I've uploaded an image of all the dice possibilities.

Example Scenario

ATTACKER'S DICE		DEFENDER'S DICE
Yellow (D12)	2	Red (D12)	1
Green (D8)	4	Purple (D8)	2
Blue (D6)	2	Black (D6)	1

Factor 1: There are multiple of each dice and each color dice has different result possibilities (Refer to image for percentages)
Factor 2: Some sides of the dice count as 1 Hit, some 2 Hits, and some 3 Hits. (And the same goes for the dice used for blocking)
Factor 3: I am comparing the number of Hits vs the number of Blocks rolled. Each Block cancels out a Hit.
Factor 4: A tie goes to the Defender because in the case all of the Hit's would be Blocked.

So the situation is (Yellow, Green, Blue used for attacker) VS (Red, Purple, Black used for defender).
I'm needing to know what th

e percentage is that the ATTACKER lands at least 1 more Hit than the DEFENDER does Blocks (Basically I need to know what the chances are that the attack will hit the defender.

The attached image shows the number of sides on the dice that count for each Hit/Block. For example: the yellow dice has 1 side that counts as 3 Hits (8% chance of landing on that side).

If you need any more information let me know! I greatly appreciate any help I can get! I've been trying to figure out how to do this for a while and can't seem to wrap my head around all the possibilities.

 

[giants756]

EDIT (Sorry this got covered by the image): I'm needing to know the percentage that the ATTACKER lands at least 1 more Hit than the DEFENDER does Blocks (Basically I need to know what the chances are that the attack will hit the defender)

 

[Snakehips]

@giants756
So are you throwing 8 Attack dice to get a total of Hits, and 4 Defence dice to get a total of Blocks?
You want the probability that Hits - Blocks > 1 ?

 

[giants756]

@giants756
So are you throwing 8 Attack dice to get a total of Hits, and 4 Defence dice to get a total of Blocks?
You want the probability that Hits - Blocks > 1 ?

You are correct on the number of Attack and Defense dice. I want the probability the there is at least 1 more Total Hits than Blocks. That may be what you were saying but wanted to make sure. Chances that Hits > Blocks.

 

[Snakehips]

@giants756 Please accept my apologies. I'm not a mathematician but fancied the challenge of taking a look at this. The upshot is that I have given it more of my limited lifespan than would appear sensible and still I cannot get my head around it. I am now both sleep deprived and deflated!

Sorry I haven't been able to help.
Good Luck!

 

[giants756]

@giants756 Please accept my apologies. I'm not a mathematician but fancied the challenge of taking a look at this. The upshot is that I have given it more of my limited lifespan than would appear sensible and still I cannot get my head around it. I am now both sleep deprived and deflated!

Sorry I haven't been able to help.
Good Luck!

It’s ok! I really appreciate you trying to help! I’ve struggled with it for a while myself. Thanks again for giving it a shot!

 

[Eric W]

I haven't quite figured out the scenario myself. It appears the attacker throws 1 12-sided die, 1 8-sided die, and 1 6-sided die (3 dice), and the defender rolls the same. Are you looking for the total score against the total score? Or do you have to handle the dice pairwise (attacker v. defender)? And I don't quite grasp the single-hit, double-hit, triple hit twist.

In any case, just assuming total score v. total score, you can figure out the probability by the basic definition of # of successes/# of cases. I wrote a quick program that goes through every possible roll and figured out the odds, and it turns out that the attacker has about a 47% chance of hitting. This seems reasonable, since the ties go to the defender.

 

[giants756]

I haven't quite figured out the scenario myself. It appears the attacker throws 1 12-sided die, 1 8-sided die, and 1 6-sided die (3 dice), and the defender rolls the same. Are you looking for the total score against the total score? Or do you have to handle the dice pairwise (attacker v. defender)? And I don't quite grasp the single-hit, double-hit, triple hit twist.

In any case, just assuming total score v. total score, you can figure out the probability by the basic definition of # of successes/# of cases. I wrote a quick program that goes through every possible roll and figured out the odds, and it turns out that the attacker has about a 47% chance of hitting. This seems reasonable, since the ties go to the defender.

Yes I agree that 47% is pretty accurate assumption because the purple dice has more blocking opportunities than the green does hits. To answer your question, yes I’m looking for total vs total.

I think I found something that gives me an accurate representation without being a percentage. It’s basically an Hit to Block ratio and it seems to be a good representation of what I’m wanting.

 

[Snakehips]

@Eric W
My understanding is that Attack has 2 x 12-sided, 4 x 8-sided and, 2 x 6-sided dice. Each dice has a 'Hit"-scoring and non-scoring sides as per @giants756 's image in post# 1
So some faces score nothing, others, 3, 2 or, 1.
Similarly, Defence has 1 x 12-sided, 2 x 8-sided and, 1 x 6-sided dice and rolls 'Blocks' that counteract attacking hits.

By my reckoning, there are 21,233,664 possible combinations from the 8 Attack dice and, 4,608 from the 4 defence dice.
The maximum possible Defensive score is 8.
The maximum possible attacking score is 16. So any attacking score of 9+ cannot be defended and guarantees at least 1 hit.
There are 65,536 ways for Attack to throw 0 ie a 0.306% probabiity.
After that.... it starts to get smelly!

 

[Snakehips]

I see that I was a little slow in responding^^^
I was just about to see if I can get a decent probability of throwing scores 0 - 8, based on the chart below which shows the required 1,2,3's needed (any balance being 0's)
Note that combinations needing more than one 3 cannot be thrown.

Book1
	N	O	P	Q	R	S	T	U	V	W
1	Score
2	0	0
3	1	1
4	2	11	2
5	3	111	12	3
6	4	1111	112	22	13
7	5	11111	1112	113	122	23
8	6	111111	11112	1113	1122	123	33
9	7	1111111	111112	11113	11122	1123	1222	133
10	8	11111111	1111112	111113	111122	11123	11222	2222	233
11	9	111111111	1111112	1111113	1111122	111123	111222	12222	1233	333
Sheet2

@giants756 What is the motive behind you wanting this? How accurate do you need it? Is it worth me missing the football on TV to pursue it?