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Say I have a UUID a9318171-2276-498c-a0d6-9d6d0dec0e84.

I then remove all the letters and dashes to get 9318171227649806960084.

What is the probability that this is unique, given a set of ID's that are generated in the same way? How does this compare to a normal set of UUID's?

JavascriptLoser
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  • I guess you mean they all have the same probability to be generated. By "the same way" you mean "same letters at the same places"? – xavier Apr 15 '19 at 23:16
  • @xavier I mean given a set of `n` UUID's that have had their letters and dashes removed, what is the probability that they are all unique (no exact matches). – JavascriptLoser Apr 15 '19 at 23:19
  • But why do you remove the letters? Two different UUIDs might have letters at different places... – xavier Apr 15 '19 at 23:21
  • @JavascriptLoser did my answer make sense? – Nathan Apr 17 '19 at 18:22

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UUIDs are represented as 32 hexadecimal (base-16) digits, displayed in 5 groups separated by hyphens. The issue with your question is that for any generated UUID we could get any valid hexadecimal number from the set of [ 0-9,A-F ] inclusive.

This leaves us with a dilemma since we don't know, beforehand, how many of the hexadecimal digits generated for each UUID would be an alpha-characte: [A-F]. The only thing that we can be certain of, is that each generated character of the UUID has a 5/16 chance of being an alpha character: [A-F]. Knowing this makes it impossible to answer this question accurately since removing the hyphens and alpha characters leaves us with variable length UUIDs for each generated UUID...

With that being said, to give you something to think about we know that each UUID is 36 characters in length, including the hyphens. So if we simplify and say, we have no hyphens, now each UUID can be only be 32 characters in length. Building on this if we further simplify and say that each of the 32 characters can only be a numeric character: [0-9] we could now give an accurate probability for uniqueness of each generated, simplified, UUID (according to our above mentioned simplifications):

Assuming a UUID is represented by 32 characters, where each character is a numerical character from the set of [0-9]. We know that we need to generate 32 numbers in order to create a valid simplified UUID. Now the chances of selecting any given number: [0-9] is 1/10. Another way to think about this is the following: each number has an equal opportunity of being generated and since there are 10 numbers: each number has a 10% chance of being generated.

Furthermore, when a number is generated, the number is generated independently of the previously generated numbers i.e. each number generated doesn't depend on the outcome of the previous number generated. Therefore, for each of the 32 numeric characters generated: each number is independent of one another and since the outcome of any number selected is a number and only a number from [0-9] we can say that each number selected is mututally exclusive to one another.

Knowing these facts we can take advantage of the Product Rule which states that the probability of the occurrence of two independent events is the product of their individual probabilities. For example, the probability of getting two heads on two coin tosses is 0.5 x 0.5 or 0.25. Therefore, the generation of two identical UUIDs would be:

1/10 * 1/10 * 1/10 * .... * 1/10 where the number of 1/10s would be 32.

Simplifying to 1/(10^32), or in general: to 1/(10^n) where n is the length of your UUID. So with all that being said the possibility of generating two unique UUIDs, given our assumptions, is infinitesimally small.

Hopefully that helps!

Nathan
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    Great answer, thank you! You've clearly outlined it wasn't possible to answer my question in it's original form, and have given me a viable alternative. – JavascriptLoser Apr 18 '19 at 21:57