Does anyone else share your four-digit PIN?

Question

Your bank PIN is a fixed 4-digit code (one of 10000 possibilities). Independently, 3000 other customers each get a random 4-digit PIN, uniform over all 10000 codes. What is the probability that at least one other customer shares your exact PIN?

Accepted Answer

You are comparing each of the 3000 other PINs against your one fixed code, so this is a specific-target match, not an any-two match. Use the complement, nobody matches you. Each other customer misses your code with probability \frac{9999}{10000}, independently, so P(	ext{no match}) = \left(\frac{9999}{10000}ight)^{3000} \approx 0.741. Therefore P(	ext{at least one shares yours}) = 1 - 0.741 \approx 0.259, about 26\%. Sanity check: the expected number of matches is 3000/10000 = 0.3, and 1 - e^{-0.3} \approx 0.259, matching the exact answer. Takeaway. With 3000 chances against a fixed 1/100…

Does anyone else share your four-digit PIN?

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