Fusilliban's Contest

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I have a list of 9,959 songs. I'm listening to it on shuffle. Some of those mp3s are duplicates, because I have downloaded all the albums available at ocremix.org. Some of those albums have both FLAC/WAV copies designed for CD burning, and mp3 copies, so there are duplicates in my selection. What are the odds that when I hit "next song," the song that comes up is the same as the one that I was already listening to?

(It turns out that I actually have three copies of some of them, because some of OCR's album pieces are also released there as normal singles and I'm pretty comprehensive about my mirroring. You may ignore this possibility for the purposes of this puzzle. You can also assume that OCR albums are my only source of duplicates, even though there are a few more floating around my actual collection, and that Winamp's shuffle is perfectly random.)

The best answer to this, recieved before the end of my show, gets a piece of Radio gear of my choosing. Ties in bestness, if any, will be broken by the "which one arrived in my kmail box first" metric. Answers I think suitably awesome, but not most awesome, will be rewarded to the tune of one (1) radio button candy. You may submit multiple answers, but any given answer is judged based on when it came in, not on when you first submitted.

Be complete. You should not need to make any estimations or assumptions beyond what I have given. Show your work. Winners don't do drugs. Rosebud was a sled.

These are the answers in the running. You're welcome to discuss them on the discussion page, because I'm not sure which math is actually right.

From Grimdel (#902119) [reply] [quoted]
Date: Monday, December 08, 2008, 05:09PM

ok -> facts:

*9959 songs
*OC remix albums are the only source of duplicates
*duplicates are because of multiple formats: wav, flac, and mp3

With this information, we can obtain the following:
*there are 12 albums at OC Remix w/ 318 different songs

*6 albums have dual formats:
tracks title f1 f2
25 chrono mp3 flac
13 doom flaq mp3
23 doom phobos mp3 flaq
45 ffvii wav mp3
11 kirby flac mp3
24 super street fither flac mp3

w/ this info, you chances of playing the same song twice is the sum of the chances of playing any of these songs twice. In the work following - the first column is the # of songs for a given album divided by the # of songs total. The second part is the chance of pulling it's duplicate out of the remaining song list.

(25/9959)*(1/9958) +
(13/9959)*(1/9958) +
(23/9959)*(1/9958) +
(45/9959)*(1/9958) +
(11/9959)*(1/9958) +
(24/9959)*(1/9958) = 141 out of 99,171,722

From Jefromi (#1706655) [reply] [quoted] Date: Monday, December 08, 2008, 05:43PM

Contest entry:

I counted the songs by downloading the torrents, letting them create the empty files, renaming their directories to a common prefix (a) and running the following script:

  1. !/bin/bash

for dir in `ls -d a*` do cd $dir echo $dir":" echo "FLAC/WAV: "`find ./ -type f -regex '.*(flac$|wav$)' | grep 'flac$|wav$' | wc -l` echo "MP3: "`find ./ -type f -regex '.*mp3$' | wc -l` cd .. done

(I also had to assume the torrent containing a RAR had no duplicates, since the info said it contained only one filetype)

There are n=9959 songs in total. Counting the ocremix album downloads, we find k=23+25+14+45+11+66= 184 pairs of duplicates. This leaves n-2k songs without duplicates and 2k songs with duplicates. The probabilities we are currently listening to a song in each class are (n-2k)/n and 2k/n, respectively. If a song has no duplicate, there is a 1/n probability of shuffle picking it again; a duplicated song has a 2/n probability of repeating. The net probability of a repeated song is therefore: ((n-2k)/n) * 1/n + (2k/n) * (2/n) = (n+2k)/n^2 = 449/4312247 ~= 1.04122*10^-4

If Winamp avoids picking the exact same song (same file), the probability is reduced to: (2k)/n * (1/n) = 2k/n^2 = 16/4312247 ~= 3.71036*10^-6

From Stevo56 (#626528) [reply] [quoted] Date: Monday, December 08, 2008, 04:49PM

Damn it, last shot at this. This time I've got it. Ok, here are the stats: 9959 tracks total 184 duplicate flac and wav files between the albums 9959/184=54.125, which means there's one duplicate for every 54 tracks. The chances of picking one of the duplicates is 1 in 54. The chances of picking a song and then its duplicate immediately after are (1/54.125)x(1/ 54.125), which is 1 in 2929.5. Therefore, the odds of picking a song and then picking its duplicate immediately after in a random draw are 1 in 2929.5, or approximately 0.034%.

Honorable Mentions:

From thisiswhatwecalltragedy (#1826598) [reply] [quoted]
Date: Monday, December 08, 2008, 05:26PM

The answer is 1/2,one half, or fifty percent. If you are struggling with this concept, please consult your local television. The chance of something worth watching being played at any given time is also 1/2, one half, or fifty percent. 
This is truly much more simple than it seems. There are not anywhere near 9,000 different possibilities. There are two: Either it will be the same song, or a different song. Both scenarios are plausible and both are common occurrences for anyone that plays music on shuffle and uses more than one copy of certain songs. Either the song will be the same, or a different song will play. 2. 
Some would say that there are other chances involved. But, that is unnecessary. You may say, There is a one percent chance that the same song will play. This is a very small chance, in comparison with 100%. And yet, it can happen regardless. If it does, where are you left? In the back of your own mind, jabbing your cerebral cortex with a plastic spork, disappointed that you did not account for the one percent chance and take action accordingly. 
In a sense, nearly all percentiles are ranked by these means. Contrary to popular belief, percents dont mean much, nor do potential ratios or likelihoods. Chances are the least meaningful of all. The clause is the cases of 0% and 100%; in these exceptions, something definitely will or will not occur. But anywhere in between, there are two possibilities: Something will happen, or something will not happen.
And thus; <b>1/2</b>.

From Shzam (#1697381) [reply] [quoted] Date: Monday, December 08, 2008, 06:19PM

i would assume that there is a 0.002 chance of a song coming up, but i think that being a religious Pastafarian and praying to the FSM would provide the best odds, as defined by His Noodly Mind. May you be blessed by his Royal Meatballs, Ramen.

Dishonorable Mentions:

From MrUuuki (#1708584) [reply] [quoted]
Date: Monday, December 08, 2008, 06:21PM

You have 9,959 songs. Some of them are duplicates. The probability that one will turn up twice is who CARES because only a hopeless loser would have to time to figure out the problem. The point is that if a song does come up twice in a row, I'm gonna be pissed off because I like my music diverse. I think most people do. Unless, of course, you're a music racist. Is that it? Playlist segregation. Wow. I loved you, man. I thought you were bigger than that. Just... wow. I hate to break it to you, but techno, jazz and rap all deserve equal rights. I'm disappointed. Very disappointed.

From Istanbul (#651875) [reply] [quoted] Date: Monday, December 08, 2008, 04:43PM

In order to determine the odds of repetition, we have to look at the total number of songs (9599) and consider that for our example - let's call it N - to be the same song, we'll have to hear it twice in a row.

Now, this leads us to 2N - we want to hear it twice, remember) - so we come out with: 2N = 9599

Thing is, that's obviously not correct. Now, ocremix.org was founded in 1999, which means that...assuming that they get X songs per year...we wind up with 10X songs.

Now we divide 9599 into that, which gives us 2N = (10X / 9599).

Of course, the real problem occurs when we consider your triplicate songs, increasing the odds by 150%. So we instead have: 3N = ((10X / 9599) / 1.5)

Of course, the site says it has over 1500 remixes right now. So, if we're going to factor that in, we wind up with: (1500 / (3N = ((10X / 9599) / 1.5))

Now, as everyone knows, X is the roman numeral for ten. So what we really have is: (1500 / (3N = ((100 / 9599) / 1.5)

From there, it's simple math. 100 / 9599 = 0.0104 / 1.5 = 0.0069. So far, we have: (1500 / (3N = 0.0069))

Divide 3 by both sides of the equal sign, and we end up with N = 0.0023. Divide that into 1500, and we get the answer.

There is a 217391.3% chance that any given song will be a repeat of the one before. Proof once again that math can solve any statistical riddle!



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