# What are the Chances?

Nov 13 2004 Sat

2:58 am PHT

I opened Winamp a while ago and set it to play my playlist consisting of 75 songs in shuffle (random) mode. Imagine my surprise (and delight) when I heard it play my three currently most tripped songs one after the other within the first five songs! What are the chances of that happening? I would’ve believed it’s a sign from God except that I don’t know what it is a sign of.

And to prove how geeky I am, I actually computed that probability. Assuming that Winamp will play a random song from the playlist after finishing the current song (which is not the case—Winamp, I think, does not repeat the last few songs), the probability that three particular songs will be played one after the other in a queue of five randomly selected songs from a pool of 75 is 0.0041%. This answer was derived from the formula below:

The first term expresses the probability that the three songs are the first songs in the queue. So the probability that the first song is any of the three is 3 out of 75. The probability that the next song is one of the two remaining is 2 out of 75. (Remember, we assumed that Winamp could repeat a song.) And finally the probability that the third is the the last remaining song is 1 out of 75.

The second term expresses the probability that the first song is *not* any of the three particular songs and that the three songs are played one after the other within the next four songs. So the probability that the first song is not among the three is 72 out of 75. Multiply this by the probability that the three songs are played one after the other within the next four songs; the derivation is similar to our reasoning thus far.

Correct? Please complain vehemently if you don’t agree with my calculations.

To give an interesting twist, the probability approaches 1 (or certainty) as the number of songs in the queue increases from 5 and approaches infinity. Intuitively, this should make sense. As you play more and more songs, the chances that the three songs will be played one after the other increases. Only when you have played an infinite number of songs can you be sure that the three songs have indeed been played one after the other.

At this point, you might want to take some paracetamol.

Oh, by the way, the three songs in question are: Kina’s “Girl from the Gutter”, Freestyle’s “Once in a Lifetime”, and Michelle Branch’s “Breathe.”

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