Random Magic Paths - is it truly random?

[alexti]

Ivan Pedroso said:

Hmmmmmm, why shouldn't it approach 3/8 ?!?

Do you talk about convergence? Like lim(percentage, N->inf)= 3/8?

Ivan Pedroso said:
Let us assume that the distribution behind the scenes is uniform. Then the observed frequencies will approach 1/8. Of cause getting a sequence that actually results in an observed frequency of exactly 1/8 for every path will be highly unlikely, but they WILL approach 1/8 as the sample grows.

not WILL, only likely. For example, for any sample size, you have positive probability of getting all picks in one path. For samples of large size the distribution function will be getting more or more condensed, which you may call "approach". But you can't provide N for some small value x such that percentage of picks concentrated in 3 paths will deviate from 3/8 by no more than x.

Ivan Pedroso said:
I mean: if you use a uniform distribution to generate some values, then the distributed values will look more and more uniform. And therefor adding the frequencies of the three highest represented paths will tend towards 3/8 (from above obviously). I concede that getting the result 3/8 in a test sample will "never" happen.

The resulting process (the percentage of picks concentrated in 3 paths) is also a random process. That means that for different samples of the same size will produce different results.

The chances of getting 3/8 are getting smaller as you increase the sample size. The root of the issue is that the more your sample size is, the more possible outcomes can happen. That makes every particular outcome less and less likely to happen.

This random process also has a certain distribution. Which will roughly look like:
Code:

---

   xxx

  xxxxxxx

 xxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxx

3/8                            1

---

If we pick the range [x1,x2] that covers 99% of possible outcomes, then we could show that lim (x1, N-> inf) = lim (x2, N-> inf) = 3/8 (well, I think we can show), which intuitively seems as "approach", but unlike convergence, this "approach" has a stochastic character, like the difference between *will* and *most probably will*

[Bummer_Duck]

Ok, the math you guys are throwing out is just beyond me. I would, however, dispute lumping all mages into the same catagory for a test, though. There could be a problem with 1 particular unit type. If I have some time this weekend, I'll try a few Drawf recruiting tests too.

Thanks for the responses!

[The_Tauren13]

Well, I must say your test game results show none of the odd distributions mentioned in this thread; they don't appear to be more concentrated into 3 paths, and certainly aren't lacking any one path. If Im less busy today I might actually run some tests myself, and we could even look at the elemental/sorcery ratio for hundreds of mages.

For your test games, let me combine the two:
F-25,A-24,W-22,E-28,S-20,D-25,N-25,B-23
the 'duck number' is 41%
the elemental/sorcery ratio is 1.0645

Looks good to me. Hopefully I can make some more tests today.

[alexti]

alexti said:
If we pick the range [x1,x2] that covers 99% of possible outcomes, then we could show that lim (x1, N-> inf) = lim (x2, N-> inf) = 3/8 (well, I think we can show)

Or maybe we can not... I feel really rusty. Does anybody know how to prove (or disprove) that lim (x2, N-> inf) = 3/8? I did few attempts, and didn't manage it

[The_Tauren13]

F-46;A-36;W-35;E-37;S-30;D-32;N-34;B-42
292 total...

Elemental: 154
Sorcery: 138

Duck number: 43%

Well, from our accumulated statistics, I would say Panther's Elemental/Sorcery ratio hypothesis may be fairly accurate.
No other conjectures show any consitancy. So much for my origional hypothesis

[RonD]

This is elementary probability.

With p=.5 and 292 trials, there is about a 16% probability that you would get 154 (or more) of one outcome. You cannot reject the null hypothesis that Pr(elemental) = Pr(sorcery).

[I used the BINOMDIST function in Excel to calculate that]