Duplicate PINs among twenty new users

Question

An app assigns each new user a random 3-digit PIN from 000 to 999, that is, one of 1000 equally likely codes, chosen independently. Twenty people sign up. What is the probability that at least two of them get the same PIN?

Accepted Answer

Chasing "at least two match" directly is messy because there are many ways it could happen. The clean move is to compute the opposite, all PINs distinct, and subtract from 1. Hand out PINs one person at a time and ask each to be different from everyone before: - Person 1: any PIN is fine, probability 1000/1000. - Person 2: must avoid 1 used code, probability 999/1000. - Person 3: must avoid 2 used codes, probability 998/1000. - and so on, down to person 20 avoiding 19 used codes, probability 981/1000. Multiplying these together: P(	ext{all distinct}) = \frac{1000}{1000}\cdot\frac{999}{1000}\…

Duplicate PINs among twenty new users

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