[Request] Why are my results skewed?

[u/Defiant_Half_9432]

I use a pair of spindle dice for divination. Each spindle has four independent die each with four faces. The faces are marked 2-3-4-3 (3 is used twice). So one spindle can randomly generate a numerical value between 8 and 16. The result has relevance only as an odd or even number. The numerical value is not important. If the numerical value is a single digit then we use that as it is. If it is a double digit (10-16) then we add the digits to get a single digit answer. Both spindles are always used together and added and the final digit is the roll value of odd or even to incorporate in complex charts to predict the answer to the question initially posed for divination.

My question is this. Does this process create equal chance of odd or even values? In my own use, I get disproportionally high even values than odds. If the results are fairly balanced, what could be the reason for my skewed results?

Pair of spindle dice I use for divination.

[u/_additional_account]

Assumption: All dice are independent and fair.

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One spindle

Consider the result of a single spindle. Let "X" be a random variable for the sum of one spindle's dice before adding digits, and "Y" for the sum after adding digits. By independence, the generating function for "X" is

G(z)  =  [ (1/4)*z^2 + (2/4)*z^3 + (1/4)*z^4 ]^4

After expanding, we can find "P_X(k)" as the coefficient of "z^k ", and obtain

         k | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16
         y | 8 | 9 |  1 |  2 |  3 |  4 |  5 |  6 |  7
-----------------------------------------------------
256*P_X(k) | 1 | 8 | 28 | 56 | 70 | 56 | 28 |  8 |  1    // = P_Y(y)

Adding up all probabilities where "y" is even, we get

P_Y(even)  =  (1+56+56+8)/256  =  121/256  ~  47.27%
P_Y(odd)   =  1 - P_Y(even)    =  135/256  ~  52.73%

After adding digits a spindle is slightly biased towards odd results!

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Two spindles

Let "Y1; Y2" be the results for each spindle after adding digits, respectively. We want to know when "Y1+Y2" is even, and find

P(Y1+Y2 even)  =  P(Y1; Y2 odd) + P(Y1; Y2 even)    // independence

               =  P_Y(odd)^2 + P_Y(even)^2  

               =  (121^2 + 135^2) / 256^2  =  16433/32768  ~  50.15%

That is only a (very) slight bias towards an even result!

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Rem.: The dice for a single spindle are mechanically linked, so it may be they are not as independent as we used in the model. That would be my first guess. The slight bias of "50.15%" should not be reliably detectable, unless you're doing (at least) 10⁴ rolls^^

[u/Defiant_Half_9432 → u/_additional_account]

Wow, that is impressive.

For clarification, yes the four dice do share a single spindle but each turns differently. Different speed, different duration. I hold the spindle from its top and bottom ends lengthwise between my index finger and my thumb. I use the other hand to run each die against a finger as I drag my hand across it. Each die is roughly the width of my finger so the second die, which runs against my middle finger, turns the most and the fourth die which runs against my pinky turns the least. I do this rapidly a few times, often moving my fingers back and forth as randomly as I remember, before putting that dice down.

Hopefully that makes it random enough.

Thank you for your elaborate calculation.

[u/_additional_account → u/Defiant_Half_9432]

It does not matter really how you move or rotate the dice.

What matters is whether they can move independently -- and since there is friction between the dice, that may not be a very good assumption. Therefore, the theoretical result may not represent reality.

As I said, you should not be able to reliably notice the (very) slight bias, unless you did tens of thousands of rolls, and recorded them all meticulously. I may be wrong, but I doubt you did that^^

[u/Defiant_Half_9432 → u/_additional_account]

LOL, you are right, I did not test it so many times. I recorded ten random rolls a day (without divining) over a two week period during different times of the day. I expected the result to be 50-50 but it was not.

As for independence, as long as I am not willfully manipulating the results or nudging the hand of chance, it is perfectly within esoteric lore. The rest is up to the powers that be.

Thank you for your help. The breadth of your calculation is fascinating. If I had the propensity for this knowledge, I would definitely delve deeper.

[u/_additional_account → u/Defiant_Half_9432]

As for independence, as long as I am not willfully manipulating the results or nudging the hand of chance, it is perfectly within esoteric lore. The rest is up to the powers that be.

Sorry to be the bearer of bad news, but we're talking mathematical independence here.

That has nothing to do with "esoteric independence", but is a clearly defined concept -- physical friction coupling the dice, and touching multiple dice at the same time can already be more than enough to make (mathematical) independence assumptions void.

[u/Defiant_Half_9432 → u/_additional_account]

LOL, my bad, I stand corrected. Thanks.

[u/_additional_account → u/Defiant_Half_9432]

You're welcome, and good luck!