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. (round to three decimal places as needed.) would it be unusual for you to like both of the songs? no yes (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 (round to three decimal places as needed.)

Explanation:

Step1: Calculate total number of ways to pick 2 songs out of 9

The number of ways to pick 2 songs out of 9 (using combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$ where $n = 9$ and $r=2$) is $C(9,2)=\frac{9!}{2!(9 - 2)!}=\frac{9\times8}{2\times 1}=36$.

Step2: Calculate number of ways to pick 2 liked songs

There are 4 liked songs. The number of ways to pick 2 liked songs out of 4 is $C(4,2)=\frac{4!}{2!(4 - 2)!}=\frac{4\times3}{2\times1}=6$. So the probability of liking both songs is $\frac{C(4,2)}{C(9,2)}=\frac{6}{36}\approx0.167$. Since this probability is greater than 0.05, it is not unusual.

Step3: Calculate number of ways to pick 2 non - liked songs

There are $9 - 4=5$ non - liked songs. The number of ways to pick 2 non - liked songs out of 5 is $C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4}{2\times1}=10$. So the probability of liking neither song is $\frac{C(5,2)}{C(9,2)}=\frac{10}{36}\approx0.278$.

Step4: Calculate probability of liking exactly one song

The number of ways to like exactly one song is the number of ways to pick 1 liked song out of 4 and 1 non - liked song out of 5. Using the multiplication principle, the number of ways is $C(4,1)\times C(5,1)=\frac{4!}{1!(4 - 1)!}\times\frac{5!}{1!(5 - 1)!}=4\times5 = 20$. So the probability of liking exactly one song is $\frac{C(4,1)\times C(5,1)}{C(9,2)}=\frac{20}{36}\approx0.556$.

Answer:

(a) 0.167, No
(b) 0.278
(c) 0.556