Random shuffling of songs on Spotify. Spotify is a music streaming service that offers both free and subscription options. Users can create playlists and choose to use Spotify’s random shuffling feature to play back the songs. When the shuffling feature was first introduced, many users complained that the algorithm was not working. For example, in a playlist consisting of 2 songs by The White Stripes, 2 by Adele, and 1 by Maroon Five, two possible random orderings of the songs are:

A = {Adele 1, Adele 2, White Stripes 1, White Stripes 2, Maroon Five}

B = {Adele 2, White Stripes 1, Maroon Five, White Stripes 2, Adele 1}

1. Find the probability that Adele 1 is selected as the first song to play from the playlist.
2. Given that Adele 1 is selected as the first song, what is the probability that Adele 2 is selected as the second song to play from the playlist?
3. Given that Adele 1 and Adele 2 are the first two songs selected, what is the probability that White Stripes 1 is selected as the third song to play from the playlist?
4. Given that Adele 1, Adele 2, and White Stripes 1 are the first three songs selected, what is the probability that White Stripes 2 is selected as the fourth song to play from the playlist?
5. Given that Adele 1, Adele 2, White Stripes 1 and White Stripes 2 are the first four songs selected, what is the probability that Maroon Five is selected as the last song to play from the playlist?
6. Find the probability of List A by multiplying the probabilities in parts a–e.
7. Many users considered List B to be random, but not List A. Demonstrate that the probability of List B is the same as the probability of List A. [Note: In response to user complaints, Spotify now uses a different random shuffling algorithm, one that prevents an outcome like List A from occurring.]

Step by step solution

01

Given information

$\begin{array}{c}{\mathrm{A}}_1=\text{ThefirstsongselectedisAdele}1\\ {\mathrm{A}}_{2,}=\text{ThesecondsongselectedisAdele}2\\ {\mathrm{W}}_1=\text{Thethirdsongselectediswhitestripes}1\\ {\mathrm{W}}_2=\text{Thefourthsongselectediswhitestripes}2\\ {\mathrm{M}}_5=\text{ThefifthsongselectedisMaroon}5\end{array}$

02

The probability that Adele 1 is selected

The probability that Adele 1 is selected song is Adele1= $\mathrm{P}\text{(}{\mathrm{A}}_1\text{)}=\frac{1}{5}=0.2$.

03

The probability that Adele 2 is selected

The probability that Adele 2 is selected song is Adele 2 $\mathrm{P}\left(\frac{{\mathrm{A}}_{2,}}{{\mathrm{A}}_1}\right)=\frac{1}{4}=0.25$.

04

The probability that White Stripes 2 is selected

The probability that Whites Stripes 2 is selected is$\mathrm{P}\left(\frac{{\mathrm{W}}_1}{{\mathrm{A}}_1\cap {\mathrm{A}}_{2,}}\right)=\frac{1}{3}=0.33$

05

The probability that White Stripes 2 is selected

The probability that Whites Stripes 2 is selected is$\mathrm{P}\left(\frac{{\mathrm{W}}_2}{{\mathrm{A}}_1\cap {\mathrm{A}}_2\cap {\mathrm{W}}_1}\right)=\frac{1}{2}=0.5$

06

The probability that Maroon Five is selected

The probability that Maroon Five is selected is$\mathrm{P}\left(\frac{{\mathrm{M}}_5}{{\mathrm{A}}_1\cap {\mathrm{A}}_2\cap {\mathrm{W}}_1\cap {\mathrm{W}}_2}\right)=\frac{1}{1}=1$

07

Find the probability for list

The probability of List A by multiplying the probabilities in parts a–e is

$\begin{array}{c}P\left(A\right)=P\text{(}{A}_1\text{)}\times P\left(\frac{A_2}{A_1}\right)\times P\left(\frac{W_1}{A_1\cap A_2}\right)\times P\left(\frac{W_2}{A_1\cap A_2\cap W_1}\right)\times P\left(\frac{M_5}{A_1\cap A_2\cap W_1\cap W_2}\right)\\ =\frac{1}{5}\times \frac{1}{4}\times \frac{1}{3}\times \frac{1}{2}\times 1\\ =0.0083\end{array}$

08

the probability for list B

Apply the same procedure for list B. Then the probability is 0.0083.

Therefore, the probability of list A is 0.0083.

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