Example:Alec's unmarked die experiment
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....
- Maybe use something like this to show distributions....
Modelling the situation
Assuming the volunteer was random We have some options:
- Model each digit (row) as the record of 45 die rolls
- This would model each row as [ilmath]\text{Bin} [/ilmath][ilmath]\left(\frac{1}{6},45\right)[/ilmath]
- 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
- ↑ Each die was a generic white black-spotted die, an individual die could not be identified from the group of 5 dice.
- ↑ 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.
- ↑ 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.
- ↑ 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)
- [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