Question

suppose that a spotify playlist has 8 tracks. after listening to all the songs, you decide that you like 3 of them. with the random feature on the playlist, each of the 8 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 8 songs are played (if, for example, track 2 can play twice in a row).
(a) the probability that you like both songs is 0.107
(round to three decimal places as needed.)
would it be unusual for you to like both of the songs? choose the correct answer.
no
yes
(b) the probability that you like neither song is
(round to three decimal places as needed.)

Explanation:

Step1: Calculate the number of non - liked songs

There are \(8\) tracks in total and \(3\) liked tracks. So the number of non - liked tracks is \(n = 8 - 3=5\).

Step2: Use the combination formula for probability (without replacement)

The probability of an event \(E\) is \(P(E)=\frac{\text{Number of favorable outcomes}}{\text{Number of total outcomes}}\).
The number of ways to choose \(2\) songs out of \(8\) is given by the permutation formula \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n = 8\) and \(r=2\), so \(P(8,2)=\frac{8!}{(8 - 2)!}=\frac{8!}{6!}=8\times7 = 56\).
The number of ways to choose \(2\) non - liked songs out of \(5\) is \(P(5,2)=\frac{5!}{(5 - 2)!}=\frac{5!}{3!}=5\times4=20\).

Step3: Calculate the probability

The probability that you like neither song is \(P=\frac{20}{56}\approx0.357\)

Answer:

\(0.357\)