Random Magic Paths - is it truly random?

[Ivan Pedroso]

alexti said:

Do you know how to proof it? I don't see any obvious one.

"the Law of Large Numbers" (or is it called something else, can't remember it) ensures that:
"The more you repeat the random process the more the empirically measured frequencies approach the values of the ''true'' underlying probabilities"
- repeatedly roll some eight-sided dies and your observed frequencies will get closer and closer to 1/8 as you go along.

alexti said:

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.

Sure... I should have used the interval: [3/8 ; 3/8+3*epsilon]
Adding up any three empirical frequencies will (very likely) result in a value that is included in the interval [3/8-3*e ; 3/8+3*e]. Adding the three largest will make the value end up in the smaller interval above.

(hehehe I could be a pedantic arse and state that the smaller interval is contained in the larger and still claim my statement to be true - but I simply just forgot about the obvious lower limit of 3/8 - made me look a bit foolish.)

I made a crude program to empirically calculate the Duck_Number: (I like MatLab )

Code:

---

clear                            %%% Clears stuff.

rand('state',sum(100*clock));    %%% Random seed gets mixed up a bit.

m=[0 0 0 0 0 0 0 0];             %%% A vektor is created:

                                 %%% The first entry could be FIRE

                                 %%% the second AIR, and so on.

N=100000                         %%% Number of mages to be generated.

for I=1:N                       

    roll=randint(1,1,8)+1;       %%% Random integer between 1 and 8.

    m(roll)=m(roll)+1;           %%% The appropriate entry (FIRE, AIR...

end                              %%% gets bumped up by one.

f=m/N                            %%% The generated frequencies are computed.

sf=sort(f);                      %%% Frequencies are sorted (ascending order).

duck=sf(8)+sf(7)+sf(6)           %%% The three highest values are added.

---

Doing a handfull of runs (N=100000) resulted in:
Duck_Number ~ 0.377
as a reference: 3/8 = 0.375