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If I had an infinite stream of items entering a queue, and I wanted ~10% of them to be marked in a special manner, would it be "correct" to roll a 10 sided die each item that passed, and if the die rolled a "1", mark the item?

What made me pause and ask this question is that I chose the number 1, therefore it is not random. I could also choose 5, or 6. The die however, being balanced, has an equal probability of landing on a 1, 5, or 6 so it seems like it wouldn't matter. I guess I want to know if choosing the number 1 vs 5, 6 would influence the outcome.

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"I wanted a 10% chance of an item entering a queue to be marked differently, could I roll a 10 sided die and mark the item if it's equal to 1?"

Yes.

And to answer your later questions, using 5 or 6 instead of 1 would not influence the outcome. It's still a 10% chance of being chosen and each "roll" is independent, so you could also theoretically change what needs to be rolled each time and it'd still be 10% probability. However, for programming and simplicity purposes, it is ideal to choose a specific number and keeping to it instead.

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You could do that or any process such that the chance of success on any given roll is 10%. You'd want to make sure that the trials/rolls are independent of one another.

As another example, you could have it so that the toy was "marked" if the first toy through the line coincided with a roll of 1, the second toy through the line coincided with a roll of 2, ... the tenth toy through coincided with a roll of 10, the eleventh toy coincided with a roll of 1, and so on. At each trial, the probability of success is 10% and the rolls are independent of one another.

What would not work is if you rolled all of your dice beforehand and THEN decided which numbers corresponded to marking the toys. That (pretty obviously) wouldn't randomly assign "marking" to toys.

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