Lets say I am spinning a bottle while sitting in a group of 10 people. I want to know the minimum number of times I would need to spin the bottle to have a 50% chance of hitting everyone in the circle at least once. How would I go about determining this minimum number of spins? Would the method to calculate this change with a very small (say 1%) or large (say 99%) chance, or with a much larger number of people (say 1000)?
I have attempted to look up some formulas but I am really not sure how to model this problem mathematically/in terms of probabilities. I know that it is impossible to determine if the percentage is set to 100% (no guarantee you'll ever hit each unique person). Hopefully this same math question hasn't been duplicated elsewhere with a different framing, the closest I could find was here.
EDIT: A little more searching and apparently this is a well defined problem... It seems searching for it in terms of dice rolls yielded much better results and based on my understanding should be equally applicable (replacing hitting a person with the bottle with getting a certain number on the die).
See here: https://en.wikipedia.org/wiki/Coupon_collector%27s_problem Will update when I think I have a real answer.