Question

Suppose a playlist on a music player consists of 100 songs, of which three are by a particular artist. Songs are played by selecting a song at random (with replacement) from the playlist. Let the random variable x represent the number of songs played until a song by this artist is played.

(a)

Explain why the probability distribution of x is not binomial.

(b)Find the following probabilities. (Round your answers to three decimal places.)

p(4)

(ii)

P(x ≤ 4)

(iii)

P(x > 4)

(iv)

P(x ≥ 4)

(c)

Interpret each of the probabilities in part (b) and explain the difference between them.

(i)

p(4)

a. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that at least four songs would be played up to and including the first song by this artist.

b. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that exactly four songs would be played up to and including the first song by this artist.

c.  If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that more than four songs would be played up to and including the first song by this artist.

d. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that less than four songs would be played up to and including the first song by this artist.

e. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that at most four songs would be played up to and including the first song by this artist.

(ii)

P(x ≤ 4)

a. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that at least four songs would be played up to and including the first song by this artist.

b. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that exactly four songs would be played up to and including the first song by this artist.     

c. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that more than four songs would be played up to and including the first song by this artist. d.

d. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that less than four songs would be played up to and including the first song by this artist

e.If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that at most four songs would be played up to and including the first song by this artist.

(iii)

P(x > 4)

a. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that at least four songs would be played up to and including the first song by this artist.

b. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that exactly four songs would be played up to and including the first song by this artist.

c. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that more than four songs would be played up to and including the first song by this artist.

d. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that less than four songs would be played up to and including the first song by this artist.

e. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that at most four songs would be played up to and including the first song by this artist.

(iv)

P(x ≥ 4)

a. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that at least four songs would be played up to and including the first song by this artist.

b. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that exactly four songs would be played up to and including the first song by this artist.   

c. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that more than four songs would be played up to and including the first song by this artist.

d. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that less than four songs would be played up to and including the first song by this artist.

e. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that at most four songs would be played up to and including the first song by this artist.

Answer 1

(a)

The probability distribution of x is not binomial, because the number of trials is not constant and given, The trials are performed until a song by this artist is played. So, the number of trails can be any value greater than 0.

probability of selecting a song by this artist = 3/100 = 0.03

X will follow Geometric distribution with parameter p = 0.03

X ~ Geometric(p = 0.03)

PMF of X is,

for k = 1, 2, 3,...

(b)

p(4) = =  0.02738

(ii)

P(x ≤ 4) = 1 - P(x > 4) = 1 - probability of not finding the artist's song in 4 trials

=   = = 0.1147072

(iii)

P(x > 4) = probability of not finding the artist's song in all 4 trials = $0.97^4$ = 0.8852928

(iv)

P(x ≥ 4) = P(x > 3) =  probability of not finding the artist's song in all 3 trials = $0.97^3$ = 0.912673

(c)

(i)

p(4)

b. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that exactly four songs would be played up to and including the first song by this artist.

(ii)

P(x ≤ 4)

e.If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that at most four songs would be played up to and including the first song by this artist.

(iii)

P(x > 4)

c. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that more than four songs would be played up to and including the first song by this artist.

(iv)

P(x ≥ 4)

a. If the process of randomly selecting songs until a song by a particular artist is played were to be repeated many times, this is the probability that at least four songs would be played up to and including the first song by this artist.