Question

suppose you just purchased a digital music player and have put 9 tracks on it. after listening to them you decide that you like 4 of the songs. with the random feature on your player, each of the 9 songs is played once in random order. find the probability that among the first two songs played (a) you like both of them. would this be unusual? (b) you like neither of them. (c) you like exactly one of them. (d) redo (a)-(c) if a song can be replayed before all 9 songs are played. (b) the probability that you like neither song is 0.278. (round to three decimal places as needed.) (c) the probability that you like exactly one song is 0.556. (round to three decimal places as needed.) (d) the probability that you like both songs is

Explanation:

Step1: Calculate probability of liking both songs without replacement

The probability of liking the first - song is $\frac{4}{9}$. After playing one liked song, there are 3 liked songs left out of 8 remaining songs. So the probability of liking the second song given the first one is liked is $\frac{3}{8}$. The probability of liking both songs is $\frac{4}{9}\times\frac{3}{8}=\frac{12}{72}=\frac{1}{6}\approx0.167$. Since $0.167 < 0.05$ (a common threshold for unusual events), it is unusual.

Step2: Calculate probability of liking neither song without replacement

The probability of not liking the first song is $\frac{9 - 4}{9}=\frac{5}{9}$. After playing one un - liked song, there are 4 un - liked songs left out of 8 remaining songs. So the probability of not liking the second song given the first one is un - liked is $\frac{4}{8}$. The probability of liking neither song is $\frac{5}{9}\times\frac{4}{8}=\frac{20}{72}\approx0.278$.

Step3: Calculate probability of liking exactly one song without replacement

There are two cases: like first, dislike second or dislike first, like second.
Case 1: Probability of liking first and disliking second is $\frac{4}{9}\times\frac{5}{8}=\frac{20}{72}$.
Case 2: Probability of disliking first and liking second is $\frac{5}{9}\times\frac{4}{8}=\frac{20}{72}$.
The probability of liking exactly one song is $\frac{20}{72}+\frac{20}{72}=\frac{40}{72}\approx0.556$.

Step4: Calculate probability of liking both songs with replacement

The probability of liking the first song is $\frac{4}{9}$, and since the song can be re - played, the probability of liking the second song is also $\frac{4}{9}$. The probability of liking both songs is $\frac{4}{9}\times\frac{4}{9}=\frac{16}{81}\approx0.198$.

Step5: Calculate probability of liking neither song with replacement

The probability of not liking the first song is $\frac{5}{9}$, and the probability of not liking the second song is also $\frac{5}{9}$. The probability of liking neither song is $\frac{5}{9}\times\frac{5}{9}=\frac{25}{81}\approx0.309$.

Step6: Calculate probability of liking exactly one song with replacement

There are two cases: like first, dislike second or dislike first, like second.
Case 1: Probability of liking first and disliking second is $\frac{4}{9}\times\frac{5}{9}=\frac{20}{81}$.
Case 2: Probability of disliking first and liking second is $\frac{5}{9}\times\frac{4}{9}=\frac{20}{81}$.
The probability of liking exactly one song is $\frac{20}{81}+\frac{20}{81}=\frac{40}{81}\approx0.494$.

Answer:

(a) The probability that you like both songs is $\frac{1}{6}\approx0.167$. Yes, it is unusual.
(b) The probability that you like neither song is $\frac{20}{72}\approx0.278$.
(c) The probability that you like exactly one song is $\frac{40}{72}\approx0.556$.
(d)

• The probability that you like both songs is $\frac{16}{81}\approx0.198$.
• The probability that you like neither song is $\frac{25}{81}\approx0.309$.
• The probability that you like exactly one song is $\frac{40}{81}\approx0.494$.