Need Math Help on mechanic

[Sir Brennen]

So, I'm tinkering with a d10 die mechanic. In it, multiple dice are rolled, and any die with a roll over a given value is discarded. The remaining dice are summed.

For example, 5d10 are rolled with values of 6, 5, 9, 5 and 9. The target value is 7. The 9's are therefore discarded, for a total of 16 for the remainder.

What I'm looking for is, how would you compute the average final sum for X d10 dice vs. Target Number Y. X could be any whole value, but for practical purposes, will rarely exceed 10, and more often be in the 2 to 5 range. Y can be any whole value between 1 and 10.

I took a stab at it, and came up with this:

Avg = (Y + 1) ÷ 2)) * (Y ÷ 10) * X

So, for Target Number of 10 on 4 dice, it would be 5.5 * 1 * 4, or 22 (the normal average of 4d10)

For a Target Number of of 7 on 4 dice, it would be 4 * 0.7 * 4, or 11.2 So a target number of 7 will yield an average roll approximately 51% of a target number 10.

Can anyone verify I'm figuring this correctly, or want to shed any deeper statistical analysis on this mechanic?

 

[starwed]

The easiest method here is just to calculate the average result from a single die. For X dice, just multiply the result by X.

There's a Y/10 chance that we keep the result, and a (10-Y)/10 chance that we discard it. We can simply ignore the second case, since that doesn't add to our total.

In the first case, it's a flat distribution between 1 and Y, so the average result is just (1+Y)/2. So the average result will be (1/2)*(1+Y)*(Y/10)*X, which is exactly the result you got.

The only thing I can see to point out is that the average is proportional to X * (Y^2+Y). So Y is the more "important" number here.

 

[Mokona]

The average roll with a "cap" of Y where the kept values are 1, 2, 3, 4, ..., Y is (Y + 1)/2 = Y + 0.5

You roll X dice. For each die there is a (10 - Y)/10 = 100% - Y*10% chance that you treat the value as zero.

Roll five (i.e. X = 5) dice. Cap = 7 (i.e. Y = 7)

Die #1: 70% chance of avg value of 4 + 30% chance of value of 0
Die #2: 70% chance of avg value of 4 + 30% chance of value of 0
etc.

Total = 5 * (70%*4 + 30%*0) = 5 * 70% * 4 = X * Y * 10% * (Y + 1)/2 = 14 avg sum

 

[Mokona]

More fun facts about such a system... (thanks to Microsoft Excel)

If you make the difficulty easier (i.e. increase Y by 1) you get a percent increase in the avg score (holding the number of dice constant) that is independent of the # of dice.

Starting Y - % increase by adding +1 to the difficulty
1 - 200%
2 - 100%
3 - 67%
4 - 50%
5 - 40%
6 - 33%
7 - 29%
8 - 25%
9 - 22%

If you add dice (skill ranks I suppose) you get a percent increase in the avg score (holding the Y difficulty constant) that is independent of the difficulty #.

Starting # of dice - % increase by adding 1 more die to your pool
1 - 100%
2 - 50%
3 - 33%
4 - 25%
5 - 20%
6 - 17%
7 - 14%
8 - 13%
9 - 11%
10 - 10%
11 - 9%

As you can see, adding 1 die to 5 dice is half as effective as adding +1 to a 5 difficulty cap. So if you have more dice than the difficulty you're always better off making the difficulty easier.

 

[QuaziquestGM]

Just come out of the closet already and admit that you play Exalted.

 

[Sir Brennen]

Thanks starwed and Mokona. Looks like my statistical skills aren't quite as rusty as I thought. The percentages are nice as well. I was running different numbers, comparing the average score of each Y vs. Y of 10. As you noted, the percentages are consistant regardless of X. (And yes, Excel is very helpful for this stuff... )

And actually, this isn't a skill resolution mechanic, but damage dice vs. an armor rating. The skill resolution of the system (not mine, helping someone with it) uses something similar, but not a dice pool. I doubt the armor rating will ever actually be below 4 or 5.

Now that I'm confident of the statistical spread, I'm trying to figure out a mechanic for armor piercing and hollow-point bullets that make either worthwhile.

For hollow-point (worse vs. armor, more damage vs. flesh) I'm thinking y (Armor) would become y - 2, but any damage which gets through is tripled.

Armor-piercing I'm less sure about. They're obviously better vs. armor, but tend to do less tramua to tissue. So for this, I'm thinking subtract one die (x - 1) but protection is worse (y + 2; or +3 maybe?). Another option would be to have damage still be (x - 1), but the player gets to keep one die which would normally be discarded. How that would work out statistically, I dunno...

Just come out of the closet already and admit that you play Exalted.

Never played it. Is this mechanic similar? I assume it uses a d10 dice pool mechanic similar to Vampire since their both White Wolf games?

 

[QuaziquestGM]

Exalted....think a game for people who love Bo9S and think that high level (low generation) pcs in V:tM just aren't powerful enough to be fun.....

D10, target number 7 or higher, blotch only on a 1 and no successes.

The game devolves quickly into rolling buckets of d10s "to hit", and then another bucket for damage.

 

[Mokona]

Using simulation software I ran these two tests for you. Depending on the most likely values for X and Y you'd want to simulate those combinations but I guessed.
[The labels are formated as Xd vs Y.]

Forecast: 5d_vs_5
Statistic Forecast values
Trials 10,000
Mean 7.5
Median 7
Mode 8
Standard Deviation 4.03
Variance 16.22
Skewness 0.3541
Kurtosis 2.83
Coeff. of Variability 0.537
Minimum 0
Maximum 23
Mean Std. Error 0.04

Forecast: 5d_vs_7
Statistic Forecast values
Trials 10,000
Mean 14
Median 14
Mode 14
Standard Deviation 5.61
Variance 31.52
Skewness 0.1363
Kurtosis 2.7
Coeff. of Variability 0.401
Minimum 0
Maximum 33
Mean Std. Error 0.06

-----

Forecast: 2d_vs_5
Statistic Forecast values
Trials 10,000
Mean 3
Median 3
Mode 0
Standard Deviation 2.55
Variance 6.51
Skewness 0.5355
Kurtosis 2.53
Coeff. of Variability 0.8506
Minimum 0
Maximum 10
Mean Std. Error 0.03

Forecast: 2d_vs_7
Statistic Forecast values
Trials 10,000
Mean 5.6
Median 6
Mode 7
Standard Deviation 3.52
Variance 12.42
Skewness 0.2156
Kurtosis 2.33
Coeff. of Variability 0.6294
Minimum 0
Maximum 14
Mean Std. Error 0.04

-----

I ran the same test for Fireball damage (sum of Xd6) for comparison.

Forecast: 2d6
Statistic Forecast values
Trials 10,000
Mean 7
Median 7
Mode 7
Standard Deviation 2.41
Variance 5.83
Skewness 0.0352
Kurtosis 2.36
Coeff. of Variability 0.3448
Minimum 2
Maximum 12
Mean Std. Error 0.02

Forecast: 3d6
Statistic Forecast values
Trials 10,000
Mean 10.5
Median 11
Mode 10
Standard Deviation 2.98
Variance 8.91
Skewness -0.002
Kurtosis 2.54
Coeff. of Variability 0.2843
Minimum 3
Maximum 18
Mean Std. Error 0.03

Forecast: 5d6
Statistic Forecast values
Trials 10,000
Mean 17.5
Median 18
Mode 18
Standard Deviation 3.81
Variance 14.52
Skewness -0.0241
Kurtosis 2.72
Coeff. of Variability 0.2178
Minimum 5
Maximum 30
Mean Std. Error 0.04

Forecast: 10d6
Statistic Forecast values
Trials 10,000
Mean 35
Median 35
Mode 37
Standard Deviation 5.45
Variance 29.72
Skewness 0.0019
Kurtosis 2.86
Coeff. of Variability 0.1558
Minimum 14
Maximum 55
Mean Std. Error 0.05

 

[Sir Brennen]

That sound you heard was my head exploding

Actually, those stats are interesting, especially the mode of 2d_vs_5 being zero.

What simulation application did you run these through, Mokona?

 

[Mokona]

What simulation application did you run these through, Mokona?

Crystal Ball