Random Magic Paths - is it truly random?

[Ivan Pedroso]

alexti said:

Curiously enough, the correct formula (I'm 99% sure ) looks very much like yours, only the sign alternates (see my earlier post, coefficients are the same, only the sign is different).

Yeah - your initial formula is correct. But I must admit that it could do with some explanations. I looked at it (briefly, I'll admit) and could not see what it all meant, or what the idea behind the factors and numbers were. And when I saw that p(i)=0 for i=8 I just disregarded the whole thing and wrote up my (wrong) ideas. (now I get it, and that p(8)=0 is perfectly fine. It's the probability of getting 20 sages in a row with eight(!) paths missing - zero of cause.)

The alternating signs are introduced when probabilities of non-independent events are added.

Same as when non-disjoint sets are unified:

U(A,B) = A + B - I(A,B)
U: unified
I: intersection (written as an upside-down "U")

With 3 sets A,B, and C you get:
U(A,B,C) = A + B + C - I(A,B) - I(A,C) - I(B,C) + I(A,B,C)

With 8:
U(all eight) = A + B + ... + H - I(all with two) + I(all with three) - I(all with four) and so on with plus and minus alternating between the groups of intersections.

One of the ones in the group called I(all with four) could be: I(A,B,C,D) or I(A,B,D,F) or ... well anyone with four letters

- - - - we'll use the above stuff now - - - -

"A" above could mean no FIRE pick in 20 sages in a row. "B" no AIR and so on.

Then U(A,B,C,D,E,F,G,H) is all the sets that can be constructed with 20 sages where any one path is missing.

P(getting one of the sets in U(A,B,C,D,E,F,G,H)) is then the probability of getting a row of 20 sages with any one path missing. But as Alexti said, if you add together P(A)+P(B)+ ...+P(H) you will NOT get P(getting one of the sets in U(A,B,C,D,E,F,G,H)).
Because P(A) is the the probability of getting 20 sages without seeing any FIRE paths. But a series of 20 with all sorcery paths will then be a part of A, B, C, and D and would thus get counted 4 times instead of only once. The alternating signs ensures that these "extra countings" gets added and deducted correctly, in order to only count the relevant contributions once. The above describes how P(getting one of the sets in U(A,B,C,D,E,F,G,H)) should be calculated.

P(A) = (7/8)^20 (so are P(B) and P(C) and so forth)
so the first part (the one with A + B + C...) is thus:
8*(7/8)^20

P( I(A,B) ) = (6/8)^20 (and so are P( I(A,C) ) and bla bla)
the re are 28 ways to make these parings, so the second part is:
- 28*(6/8)^20

The third part is ( 8!/(5!3!)=8*7*6/(3*2)=56 ways to combine three letters from the eight available):
+ 56*(5/8)^20

And so on and so on... resulting in:
((7/8)^20 * 8c7) - ((6/8)^20 * 8c6) + ((5/8^20 * 8c5) - . . . + ((1/8^20 * 8c1) = 0.4694

As stated by Alexti and misunderstood by me, but now hopefully clear to all

[BigDaddy]

That equation looks very familiar, but I've misplace my book... (not my advance statistics books with all their distributions and tests, just the basic one).

Anyway, for anyone who stumble across this with some arcane pattern and thinks it might not be random, remember that radomness is elusive even to statisticians. Understanding randomness could be likened to understanding the actual magnitude of infinite. We can use it and test it, just like infinite, but our perception of it will never be quite accurate. So, consider a random number off of an infinitely long number line. . .

[bone_daddy]

So, does this have to do with true randomness not being possible within a system containing a limited number of variables?

[BigDaddy]

A computer can never make really random numbers, a computer scientist has ensured me that that is the case as I believed it was. Instead it uses the clock to produce seemingly random numbers. The numbers themselve seem very much random, in fact, you could call them "virtually random".

True randomness is a rather complicated term, but according to my text randomness is part of anything who's probability of occurence or omission is <100%. So, I'd say that the probability of failure of a device, or the probability of throwing heads on a coin are still random, regardless of their distribution. Distributions just describe some specific random systems. So, I'd have to say no. It's just easier to feel you've grasped random with a coin flip.

Considering a random point on an infinitely long numberline is just a good way to illustrate a humans inability to truly understand certain concepts. Cosider a point on a numberline, if you take an infinitely small movement from that number to any other position on that numberline, there are an infinite number of uniques points between the first and the second, until the movement = 0.

Similarly, we make do with random models and distribution that resemble the things we can test. So, consider this, the probability of flipping heads or tails on a coin 1000 times is 9.3326E-302. IF the coin is flipped an infinite number of times, the probability of this occuring is 1.

In this example we have a demonstration of true randomness, with only two choices (heads or tails).

[Saber Cherry]

Via's CPU's produce random numbers, using special hardware. A Turing Machine cannot make random numbers, but modern computers don't have all the limitations of Turing machines (as shown by Via).

[Verjigorm]

Interesting... is this device a miniature Geiger-Muller tube with some sort of radioactive isotope in a sealed chamber (similar to the technology used to make smoke detectors)? Is there a website where I can see a press release for it--I couldn't seem to find one in my own search (although I didn't look very long).

[BigDaddy]

That IS interesting. Basically, its like programming a robot to roll a die and then read the number, but it doesn't remove the theoretical limitation (that being that a computer is unable to come up with its own random numbers, in any case because you can forecast them knowing the time/function).

A very good use of it would be to make a pregenerated table or random numbers to seed your RNG.

[Saber Cherry]

Verjigorm said:
Interesting... is this device a miniature Geiger-Muller tube with some sort of radioactive isotope in a sealed chamber (similar to the technology used to make smoke detectors)?

Hehe... no=) It's pretty simple - it uses some sort of temperature-monitoring circuitry, with random thermal fluctuations accumulating in a large buffer to be stored until a random number is needed. The chip also has a built-in hardware cryptography engine (using AES) and I think the random numbers are utilized by that, though they are also availible to software running on the computer.

Obviously, on the macro scale, temperature is not a purely random thing, or else chemistry would be a lost cause. However, if you measure temperature precisely enough, you start getting true randomness from microscopic sources, just like if you track a dust particle closely enough, you see random Brownian motion. Is Brownian motion random? Yes, because air molecules are small enough that the Heisenburg Uncertainty Principle and momentum changes from randomly emitted photons start to play a significant role.

If you want to learn more about it, try doing a search for Via's hardware encryption engine. Also, either Ace's Hardware or ViaArena did a big feature on it a year or so back.

[bone_daddy]

Ok, so although the occurence of any given number within the set of variables will be random, there will be an emergent pattern (the distribution) which is not. Is my comprehension of what you are saying correct?

Please forgive my ignorance concerning stats and probs, but I've never taken any math courses beyond basic algebra and trig. Never really needed them. I have a tendency to only learn things as I need them, unless it's something I find interesting. :-/

[BigDaddy]

It is true that a computer can give numbers that have a nearly random distribution, BUT because they use a distribution (my brother called it an MT, for I think Marsenne Twist) and a seed (from the clock) there is an element of forecasting available that is completely impossible with true random systems. I think ANSI, which regulate random numbers I guess, has determined the the MT is the new method of choice! Still sophisticated software CAN match/nearly match random distributions, but it can't come up with random numbers. Nearly matching distributions is all you have to prove, and I don't believe that it was ever speculated that computers couldn't come close to random. In a logical system such as the computer, its impossible for the PC to pull a number out of it's B***, it's programmed how to follow a distribution.

Bone_daddy, yes, random systems have their own "patterns". Such as the normal distribution, where outliers of very high numbers are explained, but most number fall close to the average.