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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.

User Ezimet
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2 Answers

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Final answer:

To calculate the probability of liking songs on a music player with 8 tracks, consider permutations of song likes and dislikes both without and with replacement. Common probability calculations include finding the chances of liking both, neither, or exactly one of the first two songs played.

Step-by-step explanation:

The scenario describes a music player with 8 tracks, of which 4 are liked by the user. When played in random order, without replacement, the following probabilities can be calculated:

Probability of liking both of the first two songs is calculated by (4/8) * (3/7) = 1/14.
This is because there are 4 liked songs out of 8 total songs for the first pick, and then 3 liked songs out of the remaining 7 for the second pick.

Probability of liking neither of the first two songs is (4/8) * (3/7) = 3/14. There are 4 unliked songs at the start, and 3 remaining unliked songs after one is played.

Probability of liking exactly one of the first two songs can occur in two ways: liking the first but not the second, or not liking the first but liking the second. This is (4/8) * (4/7) + (4/8) * (4/7) = 4/14 or 2/7.

With replacement, the probabilities change:

Liking both first songs: (4/8) * (4/8) = 1/4.

Liking neither: (4/8) * (4/8) = 1/4.

Liking exactly one: 2 * (4/8) * (4/8) = 1/2.

Note that the likelihood of an event being unusual depends on its probability. Typically, an event with a probability much less than 5% can be considered unusual.

User Setsu
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2 votes

Answer:

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%

User Nolwennig
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