Tunes for Cruising
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.
I recently burnt a CD with a selection of songs from my favourite rock bands to listen to in my car. The order of tracks, along with the bands that sing them, as burnt onto the CD are listed below (note: these songs and bands may sound similar to some more wellknown songs and bands but that's purely coincidental).
1. Sourest Thing  V3
2. I Still Haven't Heard What I'm Listening For  V3
3. Without and With Me  V3
4. Son  Pearl Honey
5. Spin the White Square  Pearl Honey
6. Better Woman  Pearl Honey
7. Last Hug  Pearl Honey
8. Dead  Pearl Honey
9. Off He Comes  Pearl Honey
10. Finding Your Religion  M.E.R.
11. Nobody Hurts  M.E.R.
When driving along listening to this fine collection of tunes, I usually put my CD player on random (ie. the order that the tracks are played does not follow any set pattern). If my CD player plays through all 11 tracks before repeating any of them, what is the probability that no two songs from the same band will be played one after the other?
Answer
The probability of playing the tracks in a random order without any two songs from the same band being played one after the other is 1/462.
As there are six Pearl Honey songs they must be played in the oddnumbered positions. The V3 and M.E.R. songs will be played in the evennumbered positions. One such example is shown below:
1. Spin the White Square  Pearl Honey
2. I Still Haven't Heard What I'm Listening For  V3
3. Off He Comes  Pearl Honey
4. Finding Your Religion  M.E.R.
5. Better Woman  Pearl Honey
6. Without and With Me  V3
7. Last Hug  Pearl Honey
8. Nobody Hurts  M.E.R.
9. Dead  Pearl Honey
10. Sourest Thing  V3
11. Son  Pearl Honey
There are six possible Pearl Honey songs that could be played first.
There are five possible V3 or M.E.R. songs that could be played second.
For the third song, there are now only five possible Pearl Honey songs that could be played.
For the fourth song, there are now only four possible V3 or M.E.R. songs that could be played.
And so on and so forth...
Following this pattern we have that there are 6*5*5*4*4*3*3*2*2*1*1 = 86,400 playlists where no two songs by the same band are played one after the other.
All up there are 11*10*9*8*7*6*5*4*3*2*1 = 39,916,800 different playlists in which the 11 tracks could be played.
So the probability that all 11 tracks will be played without any two songs from the same band being played one after the other is 86,400/39,916,800 which can be simplified to 1/462.
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Comments
jimbo
Sep 29, 2003
 Excellent puzzle. I'm glad your tastes didn't run to one less V3 track and one more Pearl Honey track. That would have made it even more interesting! Hmmm. 
jacintan
Oct 10, 2003
 This is really good, but i think it should be put in the mathematical catogary. 
snaps
Oct 10, 2003
 The question asks for a probability. That's why it's in the probability category and not math. 
einat16
Oct 30, 2003
 "purely coincidental"  LOL! good one 
(user deleted)
Jan 05, 2004
 Snaps, and any other people that love to made probability riddles I want to give you guys props, because it really takes thinking to come up with them, and that's probably why I haven't written one yet 
fg_4ever
Jan 31, 2004
 That was really HARD. But it was very interesting to see how the answer worked out. Probability was never one of my strong subjects. 
ghost09
Apr 11, 2004
 I LOVE IT! 
CGauss6180
Oct 24, 2004
 To be honest i found this the easiest question by far the other more wordy ones r harder so i agree this really is a math permutations question. 
snaps
Oct 24, 2004
 Let me state again that the question asks for the probability of something, hence the reason why it is in the probability category. 
tsimkin
Nov 26, 2004
 I agree that it belongs in the probability section. Combinatorics is frequently a part of probability. Good question, snaps! 
bbbz
Aug 10, 2008
 i had fun just figuring out the songs. "off he goes" came out when I was getting into blues, so that one stumped me at first. 
opqpop
Sep 27, 2010
 This is an easier question than it may seem.
The key is to note a nonPearl Honey songs must go in between each of the six Pearl Honey songs. Otherwise, it will be impossible to construct an order with no two consecutive songs being the same artist.
The 5 nonPearl Honey songs can be inserted in 5! ways. The 6 Pearl Honey songs can be rearranged in 6! ways. Hence there are 5! * 6! valid configurations.
The total number of configurations is 11!.
5! 6! / 11! = 1/462 
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