Random Magic Paths - is it truly random?

[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 Panther]

Graeme Dice said:
So you are telling me that the moment when a particular atom of a radioactive element will decay can be determined before it decays? Because that's what you're claiming when you claim a deterministic universe. Of course people dispute your claim, because it's known to be false.

You should at least be accurate here. The correct wording of your last sentence ought to be "Of course some people dispute your claim, because many believe it to be false."

Just because some scientists believe that radioactive decay is totally random does not make it so. I work at a nuclear weapons research facility in the field of radiation detection, and I can assure you that this question is far from settled.

In fact, I personally believe that if the means to observe a single nucleus WITHOUT interference were to ever be developed, one would likely be able to predict when a particular radioactive atom is about to decay. But Heisenberg has a theorum which makes this a very difficult thing to achieve, even in a Gedanken experiment.

Be very careful with making sweeping judgements about the universe. It will likely bite you in the end.



On the subject of random paths for your mages, I used to think it seemed biased towards elemental over sorcery. I still do, but less severe than I used to think.

I also think that my opinion of seeing bias in my mages can perhaps be traced to such things as in my current Atlantis game, where I was REALLY wanting an earth King of the Deep, and had none after about 30 mages had been recruited. I then used my first earth mage to chain summon troll kings and now I have all the earth commanders I need.

Of course, I have since bought several Kings of the Deep with earth skill at a significantly higher rate now that it is less important to me. This may just be yet another emample of Murphy's Law!

[The Panther]

All right, I just went back in this thread and added up the four cases where people posted details with a decently high total to give us some meaningful statistics. On the conjecture of a bias of elemental over sorcery, here are the totals of elemental vs. sorcery picks.

Yvelina Game 1 : 31-17
Yvelina Game 2 : 30-28
Alexti Trial 1 : 51-45
Alexti Trial 2 : 48-48
Elemental seems favored in this to the first level.

This matches many of my own earlier trials on this subject. In fact, I can't seem to rcall of a decent-sized trial where the total sorcery randoms exceeded the total elemental randoms. The Alexti Trial 2 above is the first one I have seen where they were even equal.

Random or bias? I have no idea at this point, for one can argue either way based on the sparse data.

Later today, I will add up my Atlantis King totals and post that. I know I have not lost one as yet, so it will be a fair total. However, the game is early enough that I do not have as many Kings as any of the above four cases.

[Alneyan]

Another sample if you want, rather small, but I should be able to increase the number of these mages if my neighbours leave me in peace (those are my sages in the Yarnspinners game):

Air and Blood: 4
Astral and Fire: 5
Death and Nature: 7
Water: 9
Earth: 10
Total: 51 sages, 28 in Elemancy and 23 in Sorcery.

[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.

[johan osterman]

Duck: The random generated paths use the same function for all units with randoms odf the same kind. So if a unit has any random paths it will not have more or less of a problem than any other unit using the same allocation of randoms.

Panther: All randomly generated numbers in the game are derived from a large set of pregenerated numbers. These numbers are cycled through. While it is possible that there might be a very slight weight in the numbers towards the lower or higher range of a die, it is not a weight that should be readily perceptible.

Edit: I reread my post and realised it wasn't very clear what the point of my response to panther was, so here comes the explanation. A weight of the numbers to the lower or higher half of the distribution would be by far the likeliest cause for a predomince of either elemental or sorcery. If this was the case it would also mean that all other 'random die rolls' in the game would show similar tendencies, ie the bottom half of the distribution would show up more htan the tophalf, obviously most of those 'rolls' are not quite as readily accessible to the player though.

[Alneyan]

Another test, with Spectres this time (there are 97 of them). Here are the results:
Fire:22
Air:22
Water:25
Earth:28
Astral:22
Death:31
Nature:17
Blood:27

It gives 97 elemental picks against 97 sorcery picks. Hmm, I guess I will run another time to try to get more Sorcery than elemancy.

*Grumbles* 110 for elemental picks, 86 for sorcery. The elements win once more.

[Agrajag]

I took the sceptre test as well, 99 spectres.
Air 22 (3)
Fire 24 (0)
Earth 28 (3)
Water 26 (2)
Death 19 (2)
Nature 29 (2)
Astral 26 (1)
Blood 24 (2)
The first number is the total of results, the second is how many times that path appeared twice on the same spectre.
I find it interesting that no double fires came along, but with a chance of around 4% (Im not sure I did the math right) that's quite acceptable.
Total Sorcery: 98
Total Elemental: 100
Elemental wins, but by a very small margin.
I'd say there's no bias towards Elemental, atleast according to my tests, but with all the other tests...

[Ivan Pedroso]

atul said:

Two random numbers, x1 and x2, both with uniform distribution from zero to one. Each have an expected value of 0.5. But if you're asking what's the expected value of the _greater_ of two, that's 2/3!

You a right that if I roll a number of dies (in this case eight-sided) and then only write down the largest value every time, then the average of this "highest-value-thrown" will indeed be higher than the usual 4.5 that is the average value of a standard eight-sided die. But that is (if understand it correctly) not the situation at hand.

As I see it, we are dealing with:
Some dude rolls a bunch of eight-sided dies, and then write down how many ones he got, how many twos he got and so on. He then adds the numbers of the three most common results, and divides this number with the total number of dies rolled.

An example:
100 eight-sided dies are rolled, and the following is written down:
#1 : 15
#2 : 12
#3 : 19
#4 : 10
#5 : 9
#6 : 12
#7 : 10
#8 : 13
The three highest are added (i.e. #3,#1, and #8) and we get:
19+15+13 = 47
And get (the Duck_Number): 47/100 = 0.47

The observed frequencies of the different values of the above example are:
P(x=1) : 15/100 = 0.15
P(x=2) : 12/100 = 0.12
P(x=3) : 19/100 = 0.19
P(x=4) : 10/100 = 0.10
P(x=5) : 9/100 = 0.09
P(x=6) : 12/100 = 0.12
P(x=7) : 10/100 = 0.10
P(x=8) : 13/100 = 0.13
Which are not all that close to the 1/8 = 0.125 value that where used to generate this sample.

If you increase the number of rolled dies to a much larger number than 100, then these frequencies will be closer to 1/8. (Well the probability of getting a sample using say 1.000.000.000.000.000 dies that results in frequencies that deviate greatly from 1/8 will be extremely unlikely - that is why I say that they "will approach" 1/8)

In fact you could choose any two small positive numbers, epsilon >0 and delta >0, (could be 0.00000001 and 0.000000001) and it will then be possible to find a laaaarge number N that insures that:

If N dies are rolled then the probability of getting an observed frequency that deviates from 1/8 with more than the small number epsilon, is smaller than delta.

That is:
Probability( |"observed frequency" - 1/8| > epsilon ) < delta

And then adding up the three largest observed frequencies will then result in a value that is in the interval
[3/8 - 3*epsilon ; 3/8 + 3*epsilon]
with as close to one hundred percent certainty as you want (just choose epsilon and delta to be very small)

So yes - I do in fact claim that given N = some extremely large number, then the Duck_Number will (most most most likely) be (ever ever ever so close to) 3/8.

Sorry for all this dry boring stuff

[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