Random Magic Paths - is it truly random?

[Taqwus]

No.

Think about it: it's actually impossible for the total allocation of the three most-picked paths to be anything less than 3/8. If it were, at least one of those three paths would necessarily be underrepresented (below 1/8) and at least one of the five "rarer" paths must be overrepresented (above 1/8), which is a contradiction because then they wouldn't be the three most-picked paths.

[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]

[atul]

Ivan Pedroso said:
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.

Okay, you are the undisputed king of the hill when it comes to these things, I grant you that. Unless someone like Alexti wishes to disagree, I'll leave that dispute to you. ;p

Anyway, I was mostly worried by the way some people seemed to think that 20 or so instances would be a amount enough for representative statistics. With 100 or so randoms the distribution is hardly yet uniform, I'd call a test for a lot larger sample before anyone makes any hasty decisions...

...although that disrepancy between elemental and sorcery picks seems interesting, not all of the results are statistically significant but there's a trend forming.

[The_Tauren13]

31-17
30-28
51-45
48-48
28-23
99-93
97-97
110-86
98-100
154-138

Total of all statistics posted on this thread:
Elemental: 746
Sorcery: 675

[The_Tauren13]

Using the BINOMDIST function that is only a 3% chance... make of that what you will.

[alexti]

Ivan Pedroso said:
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

Do you know how to proof it? I don't see any obvious one. Let us consider frequency of death picks.
Code:

---

Probability to roll exactly k out of n P(k,n) = C(k,n)*p0^k*(1-p0)^(n-k). (p0 = 1/8)



For simplicity, let's consider overrunning your range up. Probability of that P(m+,n) = sum[k=m..n]{P(k,n)},

where m is smallest that satisfy m/n > p0 + epsilon.

Ignoring rounding effects we can write m=a*n, where a = p0 + epsilon.

Then P(m+,n) = sum[k=m..n]{P(k,n)}

 = p0^(a*n) * sum[k=a*n..n]{C(k,n)*p0^(k-a*n)*(1-p0)^(n-k)}.



And that's where I'm getting stuck. p0^(a*n) quickly goes to 0 when n grows,

but the sum part has number of elements proportional to n,

with the dominant n! on the top, so it will grow very quickly.

Does this P(m+,n) converge to anything? And if it does, to what value?

---

I have tried to run a test program to see what is happening.
I didn't have few billion years to wait until the probability to get within the epsilon = 0.00000001 will become distinguishable from 0, so I took 0.002 as epsilon. Unfortunately, at around n=3000 my program is running out of precision of double. At that moment P(m+,n) is around 40%. Until then it was slowly going down, but the rate of descend was decrementing. So, the experiment didn't suggest any conclusion

Ivan Pedroso said:
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]

That looks wrong. You could do this if your frequencies were independent random processes. However, in our case they are dependent from each other, because the total of all frequencies is always 1. And of course, sum of three largest frequencies is always >= 3/8, but that isn't a problem.

I'm still unsure if your theorem is right or not, but your proof needs fixing.

[alexti]

To everybody who submits statistics from the real games. Please make sure that you were not conducting any particular strategy with those random mages (meaning that you sure that some of them couldn't have been killed in the battles or by assasins, remote spells etc). Also if you were buying those mages until you got some particular pick, your statistics is also invalid (because it's guaranteed that it doesn't contain more than once instance of that pick)

[Kristoffer O]

Hmm, strangest thread I've seen in a while. I have a feeling it tells us something. Perhaps not about the topic of the thread. Interesting reading anyway

[bone_daddy]

Yeah. It tells us that we're all a bunch of nerds. :-)
Not that that's a bad thing. I'm a nerd, my wife's a nerd, and hopefully my son will grow up to be nerd.