Question

A music player or music organization program can keep track of how many different artists are in a library. First note how many different artists are in your music library. Then find the probability that if 25 songs are selected at random, none will have the same artist.

Answer 1

Let there be n artists in the library.

Probability of selecting a song with a particular artist from the 25 songs is given as

= (25/1)/(n/1)

= 25/n

 

Now the probability to select the second song that belongs to the remaining artists (n – 1), other than the first song’s artist from the remaining 24 songs is given as

= (24/1)/{(n – 1)/1}

= 24/n - 1

 

In the similar way, probability of selecting 25th song that belongs to the remaining artists (n – 24), is given as 1/(n – 24).

 

Let P(n) denote the probability that no two songs are belongs to same artist in the library that is all 25 songs are different.

 

As all the 25 songs in the random selection are equally likely, therefore the probability that all 25 songs are of different artist is obtained as

= 25/n × 24/(n – 1) × … × 1/(n – 24)

= 25!/n × (n – 1) × … ×(n – 24)