Basically, an economics class exercise caused me to wonder about this basic probability question which must have been solved before.
Imagine that an unknown positive integer n is picked, and then another person counts until n. If this person counts up to k, what is the EV of n in terms of k? Or, what is the probability your next m numbers will include n?
Is there any way of estimating this? It seems intuitive to me that if you count until 1,000,000 you should have more confidence that n will not be in the next three integers than if you counted to 10. It seems that we might not have enough information to do the EV calculation, but I'm not sure. What I think might be true is that the probability we will hit n in our next m numbers is at most m/(m+n). The trouble I have is that since we don't know how n is picked, we can't say that n is distributed in any particular way. If anyone has any thoughts I'd be happy to hear.