Will anyone else draw your raffle number?

Question

You hold a raffle ticket numbered somewhere from $1$ to $1000$ . Then $500$ other tickets are drawn independently and uniformly from $1$ to $1000$ (repeats allowed).

What is the probability that at least one of those $500$ tickets matches your specific number?

Accepted Answer

The key difference from the classic birthday problem: you only care about your one number, not about any two tickets matching each other. So you compare each of the 500 draws against your fixed value. Use the complement, nobody matches you. A single other ticket misses your number with probability \frac{999}{1000}, and the 500 draws are independent, so P(	ext{no match}) = \left(\frac{999}{1000}ight)^{500} \approx 0.607. Therefore P(	ext{at least one matches you}) = 1 - 0.607 \approx 0.394, about 39\%. A quick sanity check: with 500 chances each worth 1/1000, the expected number of matches…

Will anyone else draw your raffle number?

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