You scan the rest of the questions in this assignment and decide it is time to get back to your music career. You are now in a band called The Inclusions. Each song is now defined by a single sequence of 7 chords from the set {A, B, C, D, E, F, G} with no other restrictions (for example, (A, B, G, G, G, A, D) is a song). Each band consists of 4 members each with a unique instrument (we consider the singer as having an instrument). There is always exactly 1 singer and 1 drummer. The other two members must choose an instrument from the following set: {guitar, bass, banjo, saxophone, keyboard}. Each instrument in a band must be unique. Given a set S of songs, two songs s1 and s2 ∈ S are unoriginal if they have the same 7 chords in the same order played with the same instruments. For the questions below we will make the unrealistic assumption that all songs are chosen uniformly at random from the set of all possible original songs. 1. What is the number of possible original songs? 2. One of your songs is about to break the top 100, that is, the top 100 most popular songs in the nation for a given time period. Assuming all the songs in the top 100 are chosen uniformly at random, what is the probability that all the top 100 songs are original? 3. Your song will not get into the top 100 if it is unoriginal. Assuming that each song of the top 100 is original, what is the probability that your song is original when compared to the top 100? Note: Assume that the top 100 is chosen uniformly at random but without replacement, i.e., each song is chosen randomly but each song is chosen only once.
You scan the rest of the questions in this assignment and decide it is time to
get back to your music career. You are now in a band called The Inclusions. Each song is
now defined by a single sequence of 7 chords from the set {A, B, C, D, E, F, G} with no other
restrictions (for example, (A, B, G, G, G, A, D) is a song). Each band consists of 4 members
each with a unique instrument (we consider the singer as having an instrument). There is
always exactly 1 singer and 1 drummer. The other two members must choose an instrument from the following set: {guitar, bass, banjo, saxophone, keyboard}. Each instrument in a band must be unique. Given a set S of songs, two songs s1 and s2 ∈ S are unoriginal if they have the same 7 chords in the same order played with the same instruments.
For the
questions below we will make the unrealistic assumption that all songs are chosen uniformly at random from the set of all possible original songs.
1. What is the number of possible original songs?
2. One of your songs is about to break the top 100, that is, the top 100 most popular
songs in the nation for a given time period. Assuming all the songs in the top 100
are chosen uniformly at random, what is the probability that all the top 100 songs are
original?
3. Your song will not get into the top 100 if it is unoriginal. Assuming that each song of
the top 100 is original, what is the probability that your song is original when compared
to the top 100? Note: Assume that the top 100 is chosen uniformly at random but
without replacement, i.e., each song is chosen randomly but each song is chosen only
once.
4. There is another band called The Exclusions who also have an up and coming hit song.
The record label will promote both of your songs as long as they don’t sound similar.
Two songs are similar if they have all the same chords in the same quantity (ignoring
the order of the chords). Both your song and The Exclusions’ song use exactly 3
chords. What is the probability that the two songs use the same 3 chords in the same
quantity? For example, if your band’s song was (A, A, B, C, A, B, C) then it uses 3
A’s, 2 B’s and 2 C’s. If the Exclusions song was (C, B, A, C, B, A, A) then these songs
would be considered similar because each has 3 A’s, 2 B’s and 2 C’s. If the Exclusions
song was (C, B, C, C, B, A, A) it would not be similar to your song since they use 3 C’s
and 2 A’s.
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