Question

Suppose you just purchased a digital music player and have put 13 tracks on it. After listening to them you decide that you like 3 of the songs. With the random feature on your​ player, each of the 13 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 all13 songs are played.

Answer 1

Number of ways in which two songs can be played out of 13 songs such that the songs do not repeat = 13C2 = 78

(a) No. of ways to play the songs such that you like both of them = 3C2 = 3 (selection of 2 songs out of the 3 songs you like)

Thus, required probability = 3/78 = 1/26

(b) No. of ways to play the songs such that you like neither of them = 10C2 = 45 (selection of 2 songs out of the 10 songs you do not like)

Thus, required probability = 45/78 = 15/26

(c) No. of ways to play the songs such that you like exactly one of them = 10C1 * 3C1 = 30

Thus, required probability = 30/78 = 10/26

(d) No. of ways to play two songs out of 13 songs such that songs can be replaced = 13*13 = 169

P(you like both of them) = (3*3)/169 = 9/169

P(you like neither of them) = (10*10)/169 = 100/169

P(you like exactly one of them) = {(10*3) + (3*10)}/169

= 60/169