If I want to get a perfectly random sequence of numbers in range 1 to 6, possibly very long, and roll 5 casino dice at the same time, does the order in which I read the numbers from individual dice after each roll, matter?
Repeatedly rolling just one dice and writing down each result would be one possibility to get a perfectly random sequence, but it takes a rather long time. So to speed things up I am going to use 5 dice at the same time, to get 5 random numbers for each roll.
So the question is: From the randomness theory point of view, does the order, in which I write down the 5 numbers after each roll, matter?
Suppose my 5 casino dice have different colors: red, yellow, green, blue, pink. When I roll 5 dice, they are shuffled and there the order different colors are lying on the table after each roll is random.
Does it matter for the "quality of randomness" of my sequence if I write down dice numbers always as:
Roll 5 dice at the same time
Write down 5 numbers always IN THAT ORDER: red -> 1st number, yellow -> 2nd number, green -> 3rd number, blue -> 4th number, pink -> 5th number.
Repeat from step 1
Or if I write down the number from step 2 as: yellow, green, pink, red, blue. and at the next roll as pink, red, green, yellow, blue. and so on... (the way the lie on the table from left to right)
So does it matter, whether each dice keeps its "fixed position" in the row of five across different rolls?
Your method with the coloured dices creates random sequences
In fact, any method of choosing the order of the five dices which is determined before the outcome of the dices is valid to create a iid sequence. You could (for example) choose a new order of colours after every throw and the result would still be iid.
If the order of the five dices depend on the outcome of the dices you would introduce autocorrelation.