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I'm asked to generate random serial numbers whose probability to be guessed is less than 1/1000. How to find out in which range should these numbers be ?

Parcly Taxel
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user957479
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  • Why not $[0,1000]$? All you need is a uniform choice amongst any set with more than $1000$ numbers in it. – lulu Feb 13 '18 at 12:42
  • So if I need to apply such a serial number on 1 million products per year, does it mean that I need to generate a serial number between 0 and 1 billion to keep the probability to be guessed (for one year) below 1/1000 – user957479 Feb 13 '18 at 12:51
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    Not following. If you choose an integer from $0$ to $1000$ then I have a $\frac 1{1001}$ probability of guessing it correctly. That is what you asked for. Not sure what you are asking now. – lulu Feb 13 '18 at 12:57
  • Do you want there to be less than a $\frac1{1000}$ chance of any two out of a million products having the same serial number when the numbers are chosen entirely randomly and independently? Then you need to pick from a rather large range of numbers indeed. – Arthur Feb 13 '18 at 13:09
  • Sorry, I have some difficulty to translate my mind in a clear question. I need to apply serial numbers on products and I want to keep the probability of guessing the serial number below 1/1000. I produce 1 million products per year and I assign a serial number to each product. My conclusion would be that I need to generate serial numbers in [0, 1 billion[ in order to keep the probability of guessing any serial number below 1/1000, is this correct ? – user957479 Feb 13 '18 at 13:12
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    So: you assign serial numbers to your $N$ products, I guess a number, and you want me to have less than $\frac 1{1000}$ probability of naming a serial number which you used? Then, yes, you need more than $1000\times N$ possible numbers. – lulu Feb 13 '18 at 13:16
  • Yes, that's it, thank you very much. – user957479 Feb 13 '18 at 15:26

1 Answers1

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For purposes of clarity and completeness, I'll move the comments into an answer.

The simplest model that satisfies your constraint is to pick a number between 0 and 1000 (inclusive). As there are 1001 possible choices, the chance of guess is $\frac{1}{1001}$ which is less than $ \frac{1}{1000}$.

Clearly, any set of at least 1001 distinct integers also suffices.

Martin Roberts
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  • Thanks. When I do this --- take lots of other people's comments (and perhaps some of my own) and make them an answer -- I usually mark the answer "Community Wiki", so that I don't get any points when it's marked correct, but so that OP has a chance to mark an answer as correct and "close out" the question. – John Hughes Aug 05 '18 at 12:43