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
(round to three decimal places as needed.)

Explanation:

Step1: Calculate the probability of liking the first song

The probability of liking the first song is $\frac{3}{8}$ since there are 3 liked songs out of 8 total.

Step2: Calculate the probability of liking the second song given the first is liked

After one liked song is played (without replacement), there are 2 liked songs left out of 7 remaining. So the probability is $\frac{2}{7}$.

Step3: Calculate the combined probability

Using the multiplication rule for dependent events, $P(\text{both liked})=\frac{3}{8}\times\frac{2}{7}=\frac{6}{56}=\frac{3}{28}\approx0.107$

Answer:

$0.107$