Example: there are 10 items in a collection. Getting items in the collection when you start out with none seems fairly simple, with some duplicates being acquired over time. But when you get down to the last 2 or last item in the collection, it seems to take much longer to acquire those pieces.
Is there a mathematical graph or term to describe this situation? I was thinking of using a logarithmic curve to explain it, is there a more specific term for it?
This is known as the coupon collector problem. The expected time for collecting all items is asymptotically $n\log(n)$ if you have $n$ different items. The exact formula is $nH_n$ where $H_n$ is the $n$'th harmonic number, giving an answer of 29.3 for $n=10$. Also the waiting time for each successive new item is geometric. This is easy to see, since if you have $n$ items, $k$ of which you've collected, then you have probability $p_k:=(n-k)/n$ of getting a new item on the next turn, giving you a geometric distribution. So if you want to calculate the expected waiting time for the next new item, it's $1/p_k$. For example if you have all but 2 items, then the expected waiting time to get one new item is $n/2$. It seems rather counterintuitive that the total time to collect all items is $n\log(n)$ rather than something like $n^2$ or $n^{\alpha}$ for some $\alpha>1$.
