View source for Example:Alec's unmarked die experiment

==Experiment==
A volunteer was asked to roll 5 indistinguishable<ref group="Note" name="indistinguishable"/> (6 sided) [[dice]] (at once) onto a surface and announce the outcome as a [[sequence]] of numbers<ref group="Note" name="Sequence"/>, I recorded these outcomes as a list<ref group="Note" name="eg1"/>

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===
{| class="wikitable" border="1"
|-
! rowspan="2" | Digit
! colspan="6" | Die value
! rowspan="2" | Total
|-
! style="width:3.5em;" | 1
! style="width:3.5em;" | 2
! style="width:3.5em;" | 3
! style="width:3.5em;" | 4
! style="width:3.5em;" | 5
! style="width:3.5em;" | 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
|- style="border-top:2px solid black;border-bottom:2px solid black;"
! Total [[frequency|freq.]]
| style="text-align:right;" | 34
| style="text-align:right;" | 40
| style="text-align:right;" | 43
| style="text-align:right;" | 36
| style="text-align:right;" | 34
| style="text-align:right;" | 38
! 225
|-
! Average<ref group="Note" name="Avg1"/>
| 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....
:*: <div style="display:inline;">
{| style="border-spacing:0px;border-collapse:collapse;border:2px solid black;width:10em;height:1em;"
|-
| style="width:15.1%;background-color:#FFA000;" |
| style="width:84.9%;background-color:#DADADA;" |
|}</div>
==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 [[Binomial distribution|{{M|\text{Bin} }}]]{{M|\left(\frac{1}{6},45\right)}}
# 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 {{M|\text{Bin}\left(\frac{1}{5},38\right)}}
==Results==
: {{XXX|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 {{M|X\sim\text{Bin}\left(\frac{1}{6},46\right)}} - the distribution of values for any particular digit (if random)
* {{warning|notice the {{M|46}} should be {{M|45}}}} - I miscounted and can't be bothered to work out, the difference will be fairly small
** {{M|\mathbb{P}[X\ge 16]\eq 0.002239\ (4\text{ sf})}} - '''significant'''
** {{M|\mathbb{P}[X\le 2]\eq 0.01176\ (4\text{ sf})}} - on the fence



For {{M|Y_6\sim\text{Bin}\left(\frac{1}{5},38\right)}} - the distribution of the value of 6 for any particular digit (if random)
* {{M|\mathbb{P}[Y_6\ge 16]\eq 0.001560\ (4\text{ sf})}} - '''significant'''
* {{M|\mathbb{P}[Y_6\le 2]\eq 0.01131\ (4\text{ sf})}} - on the fence

==Notes==
<references group="Note">
<ref name="Avg1">The average of a value of "1" is {{M|6.8}}, this is {{M|\frac{34}{5} }}. Notice:
* [[Binomial distribution|{{M|\text{Bin} }}]]{{M|\left(\frac{1}{5},34\right)}} is the expected distribution of the number of 1s recorded in any particular digit
** assuming each digit is independent, and random (so a {{M|\frac{1}{5} }} chance of any particular digit)</ref>
<ref name="indistinguishable">Each die was a generic white black-spotted die, an individual die could not be identified from the group of 5 dice.</ref>
<ref name="Sequence">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 {{M|(1,2,3,4,5)}}, or call {{M|(5,3,4,1,2)}} - these are distinct, but reflect the same outcome of throwing the 5 dice.</ref>
<ref name="eg1">I may record "{{M|22665}} {{M|33456}}" - this means the volunteer announced "two-two-six-six-five" for one trial, then "three-three-four-five-six" for the next trial.</ref>
</references>
==References==
<references/>