Suppose you currently have 20 songs loaded on your iPod -- five songs by the Beatles, five songs by the Rolling Stones, five by the Who, and five by Jethro Tull.

You push the random shuffle button and listen to seven songs. You are surprised that you hear songs by only two artists. You wonder: “Is the random button on the iPod working correctly?”

What does “random shuffle” mean?

Before we can address the question, we should be clear what it means for the iPod to perform a random shuffle of the songs. Suppose we represent the 20 songs by the letters

B1, B2, B3, B4, B5, R1, R2, R3, R4, R5, W1, W2, W3, W4, W5, J1, J2, J3, J4, J5

where B, R, W, J denote a song by the artists Beatles, Rolling Stones, Who, and Jethro Tull, respectively, and we use numbers to make it clear that all of the songs are distinguishable.

Here we are listening to seven songs. A shuffle is a permutation of 7 of the 20 songs.

We know that the number of possible permutations is

20 P 7 = 20 x 19 x 18 x 17 x 16 x 15 x 14 = 390,700,800

If the iPod is doing a random shuffle, then each of the 390,700,800 possible permutations of seven songs is equally likely.

Testing a model

Here we want to “test” or check the assumption of randomness of the shuffle. We call the assumption about the sampling mechanism of the iPod a “model”. Here the model is that the iPod is producing a random shuffle of 7 of the 20 songs.

A statistic

A statistic is something that we compute from our data that makes us question or doubt the model. In our story, we observe that we hear only two artists among the seven songs. If we let

X = number of artists

Here we observe X = 2. This seems to be smaller than we might expect if the iPod was indeed giving us a random shuffle of songs.

Simulating the Probability Distribution of X

The question is whether the value X = 2 is typical or unusual if the model is true, that is, the iPod is really randomly shuffling the songs. Since X represents the number of artists among the seven songs, the possible values of X are 2, 3, and 4 (X cannot be equal to 1 since there are only five songs for a given artist).

We are interested in finding the probability distribution for X for random shuffles. This is a relatively difficult distribution to find exactly. But simulation provides a simple way of approximating this distribution.

Here is how the simulation works.

Step 1. We have the computer gives us a random shuffle of 7 songs. Here is one shuffle:

W4, J2, B3, B5, R4. W3, B4

Step 2. We compute the number of artists in my shuffle. Here all four artists are represented, so X = 4.

We repeat steps 1 and 2 a large number of times, collecting the values of X.

Making a Decision

I did this simulation 1000 times and the following table gives the frequency distribution of X. The figure plots this distribution.

X	2	3	4
Count	7	299	694
Probability	7 / 1000 = 0.0007	299 / 1000 = 0.299	694 / 1000 = 0.694

This table tells us the likely and unlikely values of X if the iPod was truly giving us random shuffles of 7 songs. From the table, we see that it is most likely that X = 4 (we see four artists among the songs) and there is approximately a 30% chance that X = 3 (there are three artists represented).

Remember we observed X = 2. The chance of observing X = 2, if the iPod is truly randomly shuffling the songs, is only 0.0007. Since this probability is very small, we reject the model that the iPod is giving us random shuffles. We conclude that there is significant evidence that the iPad is not producing random shuffles of the songs.