Suppose you just purchased a digital music player and have put 8 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 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.

a) 21.43% (It is unusual to happen)

b) 21.43%

c) 57.14%

d) Probability of you liking both songs played = 25%

Probability of liking exactly one song = 50%

Step-by-step explanation:

a)

If you like 4 of the 8 songs, the probability of liking the first song is 4/8

Then, for the second song, we have 3 songs you like among 7 songs that can be played, so the probability is 3/7.

So the probability of you liking both songs played is:

P = (4/8) * (3/7) = 0.2143 = 21.43% (It is unusual to happen)

b)

If you like 4 of the 8 songs, you dislike 4 as well, so the probability of not liking the first song is 4/8.

Then, for the second song, we have 3 songs you dislike among 7 songs that can be played, so the probability is 3/7.

So the probability of you disliking both songs played is:

P = (4/8) * (3/7) = 0.2143 = 21.43%

c)

In this case you can like either the first or the second song, so we need to sum the probabilities of both cases:

Probability of liking the first song and disliking the second:

P1 = (4/8) * (4/7) = 0.2857

Probability of disliking the first song and liking the second:

P2 = (4/8) * (4/7) = 0.2857

P = P1 + P2 = 0.5714 = 57.14%

d)

If a song can be replayed, we have:

Probability of you liking both songs played:

P = (4/8) * (4/8) = 0.25 = 25%

Probability of liking the first song and disliking the second:

P1 = (4/8) * (4/8) = 0.25

Probability of disliking the first song and liking the second:

P2 = (4/8) * (4/8) = 0.25

P = P1 + P2 = 0.5 = 50%