Your number sits out a hundred rounds

Question

You pick a number from 1 to 100. Then 100 rounds are played, and each round an outcome is drawn independently and uniformly from 1 to 100. What is the probability your number never comes up in any of the 100 rounds?

Accepted Answer

Each round misses your number with probability \frac{99}{100}, and rounds are independent, so P(	ext{never comes up}) = \left(\frac{99}{100}ight)^{100} \approx 0.366, about 37\%. There is a clean reason this is close to a famous constant. Whenever you have d outcomes and play d rounds, the "never" probability is \left(1 - 	frac1dight)^{d}, which approaches 1/e \approx 0.368 as d grows. With d = 100 we are already essentially there. So the odds your number appears at least once are 1 - 1/e \approx 0.632, a bit under two-thirds. Takeaway. Playing d rounds against d equally likely outcomes…

Your number sits out a hundred rounds

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