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).
no
yes
(b) the probability that you like neither song is 0.357
(round to three decimal places as needed.)
(c) the probability that you like exactly one song is 0.536
(round to three decimal places as needed.)
(d) the probability that you like both songs is 0.141
(round to three decimal places as needed.)
the probability that you like neither song is 0.301
(round to three decimal places as needed.)
the probability that you like exactly one song is
(round to three decimal places as needed.)

Explanation:

Step1: Calculate the probability of liking both songs (part (a))

The total number of songs \(n = 8\), and the number of liked songs \(k=3\).
The probability of liking the first song is \(\frac{3}{8}\).
After playing the first song (a liked one), there are \(n - 1=7\) songs left and \(k - 1 = 2\) liked songs left.
The probability of liking the second song given the first is liked is \(\frac{2}{7}\).
By the multiplication rule for dependent events, \(P(\text{like both})=\frac{3}{8}\times\frac{2}{7}=\frac{6}{56}\approx0.107\).
Since \(0.107<0.05\), it is unusual.

Step2: Calculate the probability of liking neither song (part (b))

The number of un - liked songs is \(n-k=8 - 3=5\).
The probability of not liking the first song is \(\frac{5}{8}\).
After playing the first song (an un - liked one), there are \(n - 1 = 7\) songs left and \(n - k-1=4\) un - liked songs left.
The probability of not liking the second song given the first is not liked is \(\frac{4}{7}\).
By the multiplication rule for dependent events, \(P(\text{like neither})=\frac{5}{8}\times\frac{4}{7}=\frac{20}{56}\approx0.357\).

Step3: Calculate the probability of liking exactly one song (part (c))

There are two cases:
Case 1: Like the first song and not like the second.
\(P(\text{like first, not like second})=\frac{3}{8}\times\frac{5}{7}=\frac{15}{56}\).
Case 2: Not like the first song and like the second.
\(P(\text{not like first, like second})=\frac{5}{8}\times\frac{3}{7}=\frac{15}{56}\).
By the addition rule for mutually - exclusive events, \(P(\text{like exactly one})=\frac{15}{56}+\frac{15}{56}=\frac{30}{56}\approx0.536\).

Step4: Calculate the probability of liking exactly one song when songs can be re - played (part (d))

The total number of ways to play two songs with replacement is \(n\times n=8\times8 = 64\).
Case 1: Like the first song and not like the second.
\(P(\text{like first, not like second})=\frac{3}{8}\times\frac{5}{8}=\frac{15}{64}\).
Case 2: Not like the first song and like the second.
\(P(\text{not like first, like second})=\frac{5}{8}\times\frac{3}{8}=\frac{15}{64}\).
By the addition rule for mutually - exclusive events, \(P(\text{like exactly one})=\frac{15}{64}+\frac{15}{64}=\frac{30}{64}= 0.469\).

Answer:

(a) The probability that you like both songs is approximately \(0.107\) and it is unusual.
(b) \(0.357\)
(c) \(0.536\)
(d) \(0.469\)