Question

Suppose you just purchased a digital music player and have put 9 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 9 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 9 songs are played.

Answer 1

a) consider the first song played . The probability of its being liked is 4/9.

And for second song, we have 3 liked songs and 8 remaining songs.

Hence probability of second being liked is 3/8.

Overall probability = (4/9)*(3/8) = 1/6 = 0.167 hence it is almost unusual.

b) First song being not liked is 5/9.

And for second song being not liked is 4/8.

Hence overall probability = (5/9)*(4/8) = 5/18.

c) To like exactly one of them, it has 2 possibilities. First song may be liked and second song disliked = (4/9)*(5/8) = (5/18)

First song may be disliked and second song liked = (5/9)*(4/8) = (5/18)

Hence overall probability = (5/18)+(5/18) = 5/9

d)  a)Probability of 2 songs liked by me with replacement means in both cases i have to select a song from 4 liked songs.

Hence probability = (4/9)*(4/9) = 16/81

b) probability of both being not liked by me is = (5/9)*(5/9) = (25/81)

c) probability of exactly one being liked = (4/9)*(5/9)+(5/9)*(4/9) = 40/81