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

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 setting 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) The probability that you like both songs is 0.214. (Round to three decimal places as needed.) Would it be unusual for you to like both of the songs? No Yes (b) The probability that you like neither song is (Round to three decimal places as needed.)

Transcript

00:01 For this exercise, we are told that we have eight tracks in our music player, and we like four of the tracks.

00:08 We also told that the music player is put on random, and we're asked to find the probability that among the first two songs, for part a, we're asked for the probability that we like both of them.

00:22 So if the song order is randomly selected, and we're looking only at the first two songs, we can say the probability that we like both of these first two songs.

00:42 Is equal to the number of ways of selecting two of the songs that we like, divided by the total number of ways of selecting two songs out of the eight tracks.

01:11 So it's like this.

01:21 So this, in the numerator, we have the number of ways to select two of the four songs we like.

01:28 So there are four songs that we like, and we are choosing two of them.

01:32 So the total number of ways to select two of the four songs we like is four choose two.

01:37 And the total number of ways of selecting two songs from 8 is 8 choose 2.

01:47 This is equal to 6 over 28 or 3 over 14 or 0 .24.

02:05 Now for the probability that we like neither of them, so it's like this.

02:29 In the numerator, the number of ways of selecting two of the four songs we don't like, divided by the number of ways of selecting two songs out of 8.

02:38 It is exactly the same as part a.

02:45 4 choose 2 in the numerator and 8 choose 2 in the denominator.

02:50 So once again we have 0 .214.

02:56 And then for c we want the probability that we like exactly one of them.

03:15 Here is one way to solve this.

03:18 It's 1 minus, 1 minus the probability that we like both, minus the probability that we like neither.

03:36 1 minus these two probabilities is the probability that we like just the first or just the second song, just one out of the two, and that is 0 .572.

03:59 Or we can calculate it in a similar way to part a and b, parts a and b.

04:06 It's the number of ways of selecting one song that we like and one that we do not like.

04:12 So for the song that we like, it's 4 choose 1, and for the song we don't like, it's 4 choose 1...

Question

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