tl;dr 3h of road, 794 songs in my iTunes library, at least 20 songs repeating twice, 3 repeating thrice. Is it normal?

I am bored while driving, sometimes I listen to podcasts, sometimes I stay with
music. For a long time I have had this strange feeling that when I touch
"Shuffle" a subset of my songs are taken, and then the shuffle happens over
those. You are like "oh I really want to listen to
*Pas n'importe quel toon*" and
it simply never comes ... while you have already heard a lot of songs multiple
times, like really a lot, ... more than once.

You can find a lot of content on the Internet about how iTunes is not random... Like this question on Stack Exchange: Why isn't iTunes shuffle random?. I am not sure whether those posts dating from the beginning of the decade are still accurate.

Reverse engineering iTunes would probably be the easiest way to be sure. But one of my frustration is to not have been better at statistics, I have been reading a bit Think Stats and I would like to see if I can try to put some statistical thinking to work.

## Our facts

- 3 hours of road, 180 minutes
- Let's take a conservative 2.5 minutes per song, that is 72 songs for the trip
- 794 songs
- At least 20 songs repeating twice, 3 repeating thrice
- Have I skipped songs? Yes, not just a bit. Let's maybe triple those 72 songs, and consider we have drawn a sample of 216 songs that are making this trip's playlist

## Our big question

Can a random process generates what we have observed?

## A serious case of birthday paradox

Reasoning by analogy is often very helpful, and a friend pointed me out that this is exactly the birthday paradox.

The birthday paradox is that you have 365 possible birthday dates, with some rough edges around the 29th February. And in a group of 23 people and consider whether two people may have the same birthday, an uneducated guess says that the probability should be fairly low, when in fact it is about 50%.

Here we do not have 365 days, we have 794 different songs and not 23 guests but 216 spots on the playlist. In terms of proportion 23 / 365 = 6.3% and 216 / 794 = 27.2%

To be honest I was also willing to be a bit analytical that just programming a simulation on that and finding out that the work was already done was both a disappointment and a relief.

## A quick simulation

First the basic functions:

A bit of test would not hurt:

Here goes the funny results:

```
Matching 20 double occurences 76.42000%
Matching 3 triple occurences 32.98000%
Matching both 22.41000%
```

So well, nothing to blink about, even though it is not the first time I have the feeling of this sketchy randomness ... it seems to actually pretty likely to occur. 20% is far from a rare event.

### Seeing more

Let's try to view the frequency of such distribution.

One repetition here only describes a single occurrence of the song in the selection.

Both graphs show "songs with two repetitions" proportion, we see here that having between 15 and 25 songs repeated twice is the norm.

We could also try to see how the selection size (the size of our playlist) impacts the number duplicates.

Which gives us a projection of the relative size of the playlist to the song library size (x axis) and the proportion of unique song in the playlist (y axis). Checking our initial assumption of 216 songs over a library of 794 songs, the proportion of 27% leads to a proportion of unique songs of around 80%, about 44 positions of the playlist are likely to be repetitions of at most 22 unique songs.

# Conclusion

My initial doubt about iTunes not being random were not really deserved. Sheer randomness has a high probability 22% to have generated the data I observed. The threshold usually used admitted in that kind of test is usually at 5%, we are way above it meaning that the initial belief of randomness is plausible.

Food for thought here, this experimentation is not relevant to conclude that iTunes is random, it only states that if iTunes were random, it could have generated those data, which is not exactly the same thing.