3

Suppose every friday night you like going in a lounge bar and drink some soft drinks. Usually you spend $t$ minutes (say, $120$) there and talking to the bartender you know that there are exactly $N$ songs (say, $100$), of $3$ minutes each, in the playlist that keeps the typical atmosphere. The algorithm of the player is made such that once a song is played, it can't be played until other $n$ (say, $20$) different songs are played, then the probability of the choice of that song returns uniform as before.

What is the probability that you'll listen to the same song twice?

Honestly I do not know how to attack this problem, I saw that this kind of problems can have solutions that are "very strange" but still beautiful.

Gabrielek
  • 1,946

1 Answers1

2

Let's calculate the probability that all the songs played will be different. The probability of hearing at least one repeated song is then $1$ minus that probability.

The first $n+1$ songs will always be different because of the shuffle algorithm.

The next song cannot be any of the previous $n$ songs, but could be any one of the other $N-n$ songs, $1$ of which you have heard before. So the probability that song $n+2$ differs is $\frac{N-n-1}{N-n}$.

The song after that cannot be any of the previous $n$ songs, but could be any one of the other $N-n$ songs, $2$ of which you have heard before. So the probability that song $n+3$ differs is $\frac{N-n-2}{N-n}$.

The song after that cannot be any of the previous $n$ songs, but could be any one of the other $N-n$ songs, $3$ of which you have heard before. So the probability that song $n+4$ differs is $\frac{N-n-3}{N-n}$.

This continues until you have heard $T=\frac{t}{3}$ songs.

I'll leave you to finish this off.

Jaap Scherphuis
  • 14,744