Example:Alec's unmarked die experiment

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Experiment

A volunteer was asked to roll 5 indistinguishable[Note 1] (6 sided) dice (at once) onto a surface and announce the outcome as a sequence of numbers[Note 2], I recorded these outcomes as a list[Note 3]

Terminology:

  • "values" will be used to describe dice readings, eg a dice showing 6 is described as "value of 6"
  • "digits" will be used to describe elements of the sequence of a reading, for example digit 3 of "22655" is "6"


We wish to investigate whether or not the recording order was "random" or whether there was some sort of order to the volunteer's reading of the die.


I then tabulated (via a tally chart) the frequencies of values for each digit to obtain:

Results table

Digit Die value Total
1 2 3 4 5 6
Digit 1 7 9 6 7 6 10 45
Digit 2 4 8 13 10 8 2 45
Digit 3 7 12 8 6 7 5 45
Digit 4 5 7 11 8 9 5 45
Digit 5 11 4 5 5 4 16 45
Total freq. 34 40 43 36 34 38 225
Average[Note 4] 6.8 8.0 8.6 7.2 6.8 7.6

Notice the extreme values of:

  • value 6 occurred '16 times as the fifth and final digit of a recording
  • value 6 occurred just 2 times as digit two


Note:
  • Maybe use something like this to show distributions....

Modelling the situation

Assuming the volunteer was random We have some options:

  1. Model each digit (row) as the record of 45 die rolls
  2. Model each value (column) as being assigned to a random one of 5 digits.
    • For example, there were 38 values of six, if random we'd expect the probability of any individual 6 value being recorded in any particular digit as 1/5
    • This would mean any particular digit of the six value for example would be modelled [ilmath]\text{Bin}\left(\frac{1}{5},38\right)[/ilmath]

Results

TODO: I want to save this and move on to another task, these are just me jotting down what I have on paper for now

For [ilmath]X\sim\text{Bin}\left(\frac{1}{6},46\right)[/ilmath] - the distribution of values for any particular digit (if random)

  • Warning:notice the [ilmath]46[/ilmath] should be [ilmath]45[/ilmath] - I miscounted and can't be bothered to work out, the difference will be fairly small
    • [ilmath]\mathbb{P}[X\ge 16]\eq 0.002239\ (4\text{ sf})[/ilmath] - significant
    • [ilmath]\mathbb{P}[X\le 2]\eq 0.01176\ (4\text{ sf})[/ilmath] - on the fence


For [ilmath]Y_6\sim\text{Bin}\left(\frac{1}{5},38\right)[/ilmath] - the distribution of the value of 6 for any particular digit (if random)

  • [ilmath]\mathbb{P}[Y_6\ge 16]\eq 0.001560\ (4\text{ sf})[/ilmath] - significant
  • [ilmath]\mathbb{P}[Y_6\le 2]\eq 0.01131\ (4\text{ sf})[/ilmath] - on the fence

Notes

  1. ↑ Each die was a generic white black-spotted die, an individual die could not be identified from the group of 5 dice.
  2. ↑ A sequence is ordered, it has a 1st term and a 2nd term, a sequence made from one by swapping its 1st and 2nd terms is a distinct sequence to the original.
    • Specifically, the volunteer may call [ilmath](1,2,3,4,5)[/ilmath], or call [ilmath](5,3,4,1,2)[/ilmath] - these are distinct, but reflect the same outcome of throwing the 5 dice.
  3. ↑ I may record "[ilmath]22665[/ilmath] [ilmath]33456[/ilmath]" - this means the volunteer announced "two-two-six-six-five" for one trial, then "three-three-four-five-six" for the next trial.
  4. ↑ The average of a value of "1" is [ilmath]6.8[/ilmath], this is [ilmath]\frac{34}{5} [/ilmath]. Notice:
    • [ilmath]\text{Bin} [/ilmath][ilmath]\left(\frac{1}{5},34\right)[/ilmath] is the expected distribution of the number of 1s recorded in any particular digit
      • assuming each digit is independent, and random (so a [ilmath]\frac{1}{5} [/ilmath] chance of any particular digit)

References

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