Any Math Geeks out there that like to mess with Dice averages?

[Olgar Shiverstone]

I notice you're talking a lot about stat totals and probablilities, and not as much about distributions. (Ok, yes, it's really the same thing -- I just find it easier to read an expectation and variance than a table of probabilities.)

That's what I find really interesting -- changing your stat generation system can do wacky things to the distributions of your scores.

That's why I like point buy, myself. If you look at it from a total points perspective, the variance between players is essentially zero.

Keep it up guys. Normal distributions make my head hurt. I'll stick to Poisson processes.

 

[tarchon]

I notice you're talking a lot about stat totals and probablilities, and not as much about distributions. (Ok, yes, it's really the same thing -- I just find it easier to read an expectation and variance than a table of probabilities.)

I've posted the entire distributions for the three different cases, so if you really want the variances, it shouldn't take but a minute with Excel to work them out.
I made a graph
http://www.public.asu.edu/~tarchon/dp.jpg

 

[Nathan]

Re: Re: Re: Re: Re: Re: Wow!!! I need to take some math courses.

What's the sum of your totals for each?

What do you mean by that? Do you mean the number of total possibilities, i.e. the number of all rolls in the (6^4)^6 rolls that generate not useless characters?

 

[Nathan]

I'm not sure if I'm interpreting this right. How do T, L, and R work? Do you have to run every permutation of those three variables through the formula and sum all of them. Does this formula work for the more simple case (n=2, k=1)? With any set of fixed n and k values, you should be able to come up with a formula only based on s. I can't see how to work that out using the formula. Can you go into a simple example, like E(s,2,1)? Brackets would also help Thanks!

The problem is that I have no sum symbol (the greek letter "Sigma") on my keyboard. With it, it would have been easier for me to post the formula.

And now to your question:

1) Yes, you sum up over all possibilities of the triple (t, l, r).
However, I made a MISTAKE! One has to sum t from 1 to s, not to k, which doesn't make much sense.

2) After summing up everything you divide! IMPORTANT: I made a second mistake when I posted the formula. At the end you have to divide by the number of all possible rolls which is s^n and not s^k! Please correct the formula in this way.

3) Why do you have problems with fixed n and k? Just plug them into the formula and you get a new one depending only on s. Of course it is worthy to ask if the new formula can be further simplified.

4) MORE Brackets: Put a big "(" after the word "of" and a big ")" before the word "over". The precendence of the operators "+", "-", "*", "/", "^" is the usual one, i.e. "^" binds more the "*" and "/", "*", "/" bind more than "+", "-". Otherwise, everything is to be read from left to right.

5) Example: n = 2, k = 1 (i.e. better die of two rolls):

We have to sum t from 1 to s, l from 0 to 1 and r from 1 to 1, i.e. is fixed and no summing needed.

Plug these values (n = 2, k = 1, r = 1) in the big summand and you will get
a formula without n, k and r which is to be summed over t from 1 to s and over l from 0 to 1 (which you can write with an ordinary "+" sign).
Please note three things: 0! = 1 by definition, x^0 = 1 by definition and
1+2+3+4+5+6+...+x = x*(x+1)/2 which can be of use for the case E(2,1,s).

Evaluating the formula, I'll get:
E(2, 1, s) = (4*s^3+3*s^2-s)/(6*s^2).
(I summed up analytically.)
For example E(2, 1, 6)
= 4,47...

If you are still interested in the topic I will explain this in further detail.

 

[Elric]

Thanks. You had me confused because dividing by s^k didn't look right and summing t from 1-->k made me think that I had made a math error in running the formula (thus the comment about the brackets). Summing t from 1-->s definitely makes it a function of s. From the one example that I've tried so far, it looks right. Just wondering, how did you come up with this formula? What kind of math does it require?

 

[Nathan]

Thanks. You had me confused because dividing by s^k didn't look right and summing t from 1-->k made me think that I had made a math error in running the formula (thus the comment about the brackets). Summing t from 1-->s definitely makes it a function of s. From the one example that I've tried so far, it looks right. Just wondering, how did you come up with this formula? What kind of math does it require?

I am very sorry about me having mistyped the formula when I posted it for the first time and thus confusing you.

You are asking what kind of math is required... Hmm. I think the most important thing to deduce such a formula is your own brain and some logic. On the other hand, it is of course important to have some knowledge of the integers and basic probability theory (Laplace's notion of probability). And, one should have a good practice in manipulating formulas.

I can recommend the following book:

Graham, Knuth, Patashnik: "Concrete Mathematics".

If you have read (and worked with) this book, you should have developed a good skill manipulating numbers, formulas and deducing your own ones.
Or, better: study maths, as I did

Now let's get a little bit more specific:

And now some ideas I had developing the formula:

Fix n, k, s. The experiment is rolling a die n times.

How many possible outcomings does this experiment have? Answer: For every roll of a die, there are s outcomings. Therefore, we have s^n = s * s * ... * s (n times) in total.

The next step is to divide the set of all outcomings. For every outcoming of our experiment lets us denote by t the value of the lowest die among the highest k ones in all n dice. Okay?
Of course, t can have the following values: 1, 2, ..., s.

Example (n = 5, k = 3, s = 6):
Rolls: 1, 5, 6, 5, 2,: -> corresponding t is 5.
Rolls: 4, 2, 5, 1, 1 : -> corresponding t is 2.

Therefore we have divided the set of all results of our experiment (rolling n die) into s subsets parametrised by t.

This will lead to summation over t in my above formula.

Have you understood everything up to now?

 

[Elric]

I'm with you so far. Thanks for the explanations!

 

[Nathan]

After having divided our big set of s^n elements into s subsets indexed by t we divide these subsets two further times:

For every outcoming let us denote by l the number of dice that show a result less than t and let us denote by r the number of dice which are among the highest k and show exactly t.
Is it clear why l has to be a number between 0 and n - k and r between 1 and k?

For example (n = 6, k = 3, s = 6):

3, 5, 2, 6, 6, 3

Here, t is 5. l is 3 and r = 1.

Next example:

4, 5, 5, 4, 6, 6

Here, t is again 5, l is 2 and r is 1.

Last example:

2, 3, 3, 3, 3, 4

Here, t is 3, l is 1 and r is 3.

Okay?