I have a playlist in iTunes that is auto-updated and has the $n$ least-frequently played songs played in alphabetical order. Assuming a uniform distribution of song titles¹, in both the playlist and the songs that will be added to it, what is the expected number of songs that will be played?
A constraint: if song $n$ is Foxtrot and song $n+1$ is Hotel and the song Golf is added after Foxtrot is played, the player goes to Hotel and not Golf.
Also, if the playlist starts out uniformly distributed, will it still be uniformly distributed when it finishes? If not, what is the shape of the distribution as the number of repeated plays goes to infinity?
- In my actual playlist, this ends up not being true, if only because of things like multi-movement classical pieces where there's an extended sequence of songs added (in reverse alphabetical order) that are sequentially organized alphabetically.