Not too long ago, several of us were betting how unlikely it was that someone would compose some of the playlists submitted to the site. There is a virtually unlimited number of songs out there, millions and millions of songs and growing everyday. So, with such great diversity and taking into account that we could chose any of the existing songs to include a playlist, we were wondering what the odds are of someone composing a given playlist. We quickly realized that the odds were too small to distinguish from nil if we considered playlists over all of the millions of possible songs, and thinking about how to factor in the relative popularity of songs made our heads hurt.

So we finally agreed to just restrict the problem to deciding the odds that the songs on a particular playlist would occur in a set of random sequences of the songs actually played by our members. Then we started debating whether we should assume a single sequence of songs (what we call a playstream) could only include a single play of each song, or if we could allow playstreams that included repeated plays of songs. Not long after that we decided this problem was really not that interesting, and decided to go to the Cantina instead. Before we left, we asked our Chief Scientist what he thought the odds were, so we could decide who lost the bet and would have to buy the drinks.

He started mumbling to himself about urn problems, multivariate hypergeomtric distributions, reduction to the binomial, and Stirling’s Approximation, and we saw he was going to turn this into a major project, so we decided to leave for the Cantina. When we got back he announced he could give us an approximation to the answer for the first case in which playstreams couldn’t include repeats, but that he still had some “calculator work” to do for the case in which playstreams with could include repeats. We of course had long forgotten the question, but thought we should humor him and pretend to still be interested. He started to explain the mathematics of the problem to us: “Assume we have a population of L songs, and that we select a set of K playstreams from our members of some average length M, what is the odds that at least one playstream includes a particular playlist of N≤M songs. We also assume …”. We decided to just post the analysis for you to read instead. (The short answer for the case without repeated songs: About 10^15-to-1, or a million-billion to one, against someone composing a particular playlist from the songs our members actually play, without considering the relative popularity of songs. No one would have won the bet, so we decided we had done the right thing by going to the Cantina when we did.)

What we hope this illustrates is how we understand and appreciate the uniqueness of each playlist you share with the community. When you compose songs into a playlist, you are capturing a feeling or making a statement that you want to share with the world. We recognize how fortunate we all are that you have chosen to share that feeling or message. We will continue to work to provide you with creative and enjoyable ways to express yourself with and about music; we hope you’ll continue to share your taste in music.