10 ms·

What does randomness look like?

474 points by cyrusradfar 14y ago

thaumaturgy 14y ago
I would love to see more like this on HN.
- cyrusradfar 14y ago
  The author is @aatishb (https://twitter.com/aatishb https://twitter.com/aatishb) on Twitter. I've followed him for some time and he's a enjoyable science/math blogger if you're interested in that.
  - thaumaturgy 14y ago
    Thank you! I've added him to my reading list.
suby 14y ago
"The one on the left, with the clumps, strands, voids, and filaments is the array that was plotted at random, like stars." Are stars really plotted at random?
- davidcuddeback 14y ago
  That's a good question. There's obviously more stars in the directions of the galactic plane (what we observe as the Milky Way), but the individual stars that we see in the night sky tend to only be the ones that are really close to us (relatively speaking). So it's a question of which effect dominates the other. I think it'd be interesting to see how well the distribution of stars fits a Poisson distribution.
  - morsch 14y ago
    Imagine you were one of the dots on the plane in the dot illustrations. Would the "constellation" of all the other dots from your position be Poisson distributed?
- pixl97 14y ago
  http://en.wikipedia.org/wiki/Observable_universe#End_of_Greatness http://en.wikipedia.org/wiki/Observable_universe#End_of_Grea... Up to the largest scales the universe is randomly clumped/stringed/voided.
- Devilboy 14y ago
  Well no, if you look at the whole sky then obviously you'd see more stars in the direction of the galactic centre (the milky way) than in the opposite direction, but if you consider only a small sliver of sky the results would be pretty close to being truly random.
gluegeorge 14y ago
Were the bombs actually mostly random?
- deleted 14y ago
  [deleted]
- rosser 14y ago
  Without asking the folks aiming them we can never be sure, but the results of the survey of South London detonations mentioned in the article was shockingly close to the Poisson distribution. So it seems likely.
- davidvaughan 14y ago
  No, they were aimed but it was a matter of trial and error. To get the data about where they'd landed, some had radio transmitters. They sent a message back to armourers in France (I assume some were deliberately non-explosive). Meanwhile German agents in England were also observing the success or failure of the V2s, and in particular where they'd landed. However, in an effort to deceive the Germans, the British started reporting the correct time of successful attacks, while mentioning an incorrect location. Moreover, a double agent called Eddie Chapman also fed false information back to the Germans. As a result, the aimers never really got a grip on ranging accurately. The bombs started landing to the south east of London. There's considerably more detail about this in Most Secret War by R V Jones, who was involved in all sorts of ruses to confuse the enemy. Well worth a read. Eddie Chapman (Agent ZigZag) was played by Christopher Plummer in the film Triple Cross. It seems to be on YouTube.
  - andrewcooke 14y ago
    does that book discuss whether / how the misinformation was used to select the south east? i suspect that was a poorer area of london and i vaguely remember some kind of scandal about the relative suffering of various parts of london and class, etc. so i wonder to what extent the south east was chosen (by the british) as a target?
- InclinedPlane 14y ago
pm90 14y ago
Very interesting! So, the basic point being made is that if you know that a set of events are random and independent and you know their mean value, then you can predict their spread? (or aggregation) edit: Hmm...another question that comes to mind: is the converse true? If the spread of values of these events do not match the poisson distribution, then can we presume them to be nonrandom? Or nonindependent? Or both?
- quantumet 14y ago
  The Poisson distribution is just one random number distribution; there are several others for situations where the events are correlated, or have other properties. Half the fun of probability is figuring out which distribution is the right one to apply to the question at hand. So if your measurements don't match up to Poisson, it doesn't mean they're not random - they could just be interdependent. So yes, for a Poisson process, the spread (standard deviation) is equal to the square root of the mean; as the number of events gets large, the Poisson distribution approaches the normal distribution, but the relationship between the standard deviation and the mean continues to hold.
- Persephone404 14y ago
  If you have enough data and to determine the distribution and it's characteristics (eg mean and s.d. for normal etc) then you can use it to predict spread.
tdyo 14y ago
This is exactly how they determine species distributions in ecology as well - http://en.wikipedia.org/wiki/Species_distribution http://en.wikipedia.org/wiki/Species_distribution Uniform dispersion would suggest some territoriality aspect of the species, and clumped dispersion would suggest a heterogeneity of resources (or any other hypothesis that could then be tested).
- aatish 14y ago
  Nice! That's a really interesting and relevant example.
henrikschroder 14y ago
Back in the day I was in a raiding guild in World of Warcraft. And back then, each boss monster in a raid dungeon had a loot table, and when you killed it, your raid got a few pieces of random loot. And I don't know how many times I, and the other guy in the guild who understood statistics, had to explain to the others that random doesn't mean uniform. If a certain piece of armour has a 1/X chance to drop from a boss, what people think should happen is that if they kill that boss X times, they should see it drop once. But the reality was of course that loot was very non-uniform. Some pieces we saw lots of times, and other pieces very rarely, despite them having the same drop chance. And the players who wanted those pieces that happened to be rare for our guild, got very, very angry. We saw the same things on the official message boards, players were furious after having spent a year killing the same raid boss once a week, and never seeing a certain piece drop for them. But simple math shows that with million of players, tens of thousands of raiding guilds, some of those will see very streaky results. These days in World of Warcraft, boss monsters drop tokens instead, and when you have X tokens, you can exchange that for a piece of armour, or a weapon, guaranteed. And noone complains about the random loot anymore.
- thurn 14y ago
  This is actually an important lesson for game designers. Real randomness is very frustrating for players! Games should be designed to not be random. For example, there's an expansion to The Settlers of Catan that lets you use cards instead of dice to ensure a nice smooth distribution of resources... I think it's a lot more fun!
  - henrikschroder 14y ago
    Oh yes. Our intuition is that random is fair and uniform, but it's absolutely not. I was bit by another random quirk in World of Warcraft. There was a long-running achievement called "What a Strange Long Journey it has Been", which took at least a year in real-time to complete, and you had to actively play during each of the ten or so in-game festivals and holidays, and do some quests and tasks during each. If you missed a festival, you had to wait another year to do it again, so if you were aiming for the achievement, you really wanted to do it in one go. During their version of Valentine's, each player got a bag of heart-shaped candy, and the task you had to complete was to pull out at least one each of the eight different heart candies. But you could only pull a piece of candy once every hour or so, each time you pulled a piece you had a 1/8 chance to get a certain piece, but the holiday was time-limited, so you had about two weeks to complete it. Sounds easy and fair, right? 1/8 chance, get all eight pieces, two whole weeks, easy! Except that there was one piece that I just never got. The piece that said "I LOVE YOU!". And as the time went by, I got more and more frantic, logged in more often so as not to miss any opportunity to pull a piece of candy, but no luck. So, I did a quick bit of math. You can pull one piece every hour for two weeks, but sleeping, not playing, missing days etc, meant that I effectively pulled ~100 pieces. The chance of missing a certain piece is 7/8, so missing a certain piece 100 times in a row is (7/8)^100, which is roughly a little bit more than one in a million. With ten million players all doing the same thing, there's going to be a number of them that will hit that "one in a million" chance, which means that whoever designed that part of the achievement didn't do their homework, didn't do the math. Because intuition tells you that 1/8 chance is plenty! Hundreds of tries, of course everyone will get all pieces! Except proper math tells you otherwise, and I was one of the "lucky" outliers. (Later, they apologized and retroactively removed that part of the achievement, so I got my purple nether-drake mount without having to wait a year extra.)
marvin 14y ago
Regarding the bomb plot at the very bottom. Examining the number of bombs dropped in each square and comparing it to the Poisson distribution, it appears that the distribution of the bombs is random. But looking at the plot on the map, it appears that the higher incidence of bombs is focused on a specific region. The Poisson distribution doesn't account for the fact that a lot of the squares with a high incidence of bombs are adjacent to each other. From my layman's understanding, it appears that the bombs were in fact targeted on a specific area, but that there was a random offset from this area regarding where the bombs actually landed. Because of this, you'd see randomness in the distribution. But the distribution of bombs wasn't really perfectly random. Is the author deliberately avoiding this point, or is there something I've misunderstood?
- patrickod 14y ago
  I believe the table at the end deals with a subsection of the 3d graph, but I could be wrong.
- feral 14y ago
  If you read the linked post its clearer whats going on: http://madvis.blogspot.ie/2010/09/flying-bombs-on-london-summer-of-1944.html http://madvis.blogspot.ie/2010/09/flying-bombs-on-london-sum... The poisson analysis is only done on a subset of the area thats in the 3d plots: "Clarke's analysis was focused on the central area of higher density here and with finer geographic coordinates. Within that area his analysis found no evidence of clustering that cannot be accounted for by a Poisson process."
  - jerf 14y ago
    Yes, it could have been a bit more clear in how they "weren't being aimed". Obviously the Germans did not just point them straight up and light them off, hoping for the best. What this really demonstrated was something more like where the noise floor for the bomb distribution was. They could hit "London" but they couldn't hit any particular place in it. Very important for the military to know that information.
- aatish 14y ago
y4m4 14y ago
Such a great article! - something nice at the HN top after a long time.
saosebastiao 14y ago
TL;DR http://www.google.com/finance http://www.google.com/finance
Zenst 14y ago
True randomness is a system that say generates a 10 digit number and 11111111 has the same odd's as appearing as the other permutations. But then thats not strictly true as we all know any system that monitors the outcome (a good example will be fruit machines) to make sure there is a even distribution of all permutation will in essence remove that level of definition we like to think as truely random. With that it gets hard to truely say what is random or what is a as yet unknown pattern. This is why many have taken the approach of not having a single source of random numbers but use many and average out from there. There again is that random as the chances with such an approach of getting a high or low value would be biased out. So with that I postulate one mans random string is another mans non-random string. So with that I define randomness as a yet undertermind sequence or a data. So the included Dilbert post is with that extreemly clever and totaly true.
- Steko 14y ago
  "hard to truely say what is random or what is a as yet unknown pattern." No, it's just saying that it's much easier to get pseudorandomness out of a computer than true randomness. "This is why many have taken the approach of not having a single source of random numbers but use many and average out from there. There again is that random as the chances with such an approach of getting a high or low value would be biased out." Technically speaking you wouldn't average you'd add them together and take the decimal portion (modulo 1). That can negate bias as long as one of the sources is good even if you don't know which one. Of course you can remove bias from a single source by Von Neumann's method although this might be computationally harder than the above: http://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_biased_coin http://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_b...
- Dylan16807 14y ago
  There is actually no such thing as a random or not random number. There are only random and not random generators. Whether a fruit machine is random depends on how is defines/ensures evenness.
laureny 14y ago
Here is an article on how to assess whether your data sample is "random enough": http://goo.gl/opl52 http://goo.gl/opl52
michaelochurch 14y ago
One under-stated but awesome thing about Poisson distributions: fishing has a Poisson distribution. (Poisson is French for "fish".)
- endtime 14y ago
  I think that would only be true in an infinite lake.
Uhhrrr 14y ago
"Gravity's Rainbow", by Thomas Pynchon, has an extended section about the Poisson distribution as applied to falling bombs, maternity wards, etc.: http://books.google.com/books?id=GGPm4I3BbxAC&pg=PT194&lpg=PT194&dq=poisson+distribution+gravitys+rainbow&source=bl&ots=APsLuTKC1F&sig=wUaE1csdIbr744iR-_ZvRVQ1sCo&hl=en&sa=X&ei=KBfVULfTI8OFiAKT04CgDw&ved=0CGIQ6AEwBQ http://books.google.com/books?id=GGPm4I3BbxAC&pg=PT194&#... (possibly NSFW)
- krymise 14y ago
  That book is exactly what I thought of when reading the article. Everyday I learn something new that seems to unlock a section of that novel that was previously opaque.
baddox 14y ago
I'm a bit confused. What is this notion of "pure randomness" that this article and many comments seem to be eluding to? Perhaps I'm just not using the same definitions of terms, but I thought a uniform distribution is still considered random.
- lutusp 14y ago
  > Perhaps I'm just not using the same definitions of terms, but I thought a uniform distribution is still considered random. No, a uniform distribution is not evidence of randomness. Consider the digits 0 - 9 repeated endlessly: 0123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789 ... Uniformly distributed? Yes. Random? No. > What is this notion of "pure randomness" that this article and many comments seem to be eluding [sic] to? First, s/eluding/alluding/ Second, although the topic is complex, one test of randomness is that an ideal compression method, one able to find and exploit any repetitive pattern, cannot compress a random sequence. Third, the term "entropy" as used in information theory is tied to randomness, as explained here: http://en.wikipedia.org/wiki/Entropy_(information_theory) http://en.wikipedia.org/wiki/Entropy_(information_theory) A quote: "The entropy rate for a [fair coin] toss is one bit per toss. However, if the coin is not fair, then the uncertainty, and hence the entropy rate, is lower." Based on that, high entropy -> high randomness. Not to oversimplify a complex topic.
  - pronoiac 14y ago
    Could you edit that long line somehow? It makes reading harder in my phone.
  - endtime 14y ago
    >Uniformly distributed? Yes. Random? No. I think you're fudging what is supposed to be uniform here. In your example, the unigrams (i.e. single digits) may be uniform, but the n-grams for n > 2 are not.
    - lutusp 14y ago
      Actually, the n-grams are periodic and repetitive also. They certainly aren't random.
tome 14y ago
Strange terminology. In mathematical parlance the arrangement of the glowworms is still described as "random", but the positions of the worms simply are not independent of one another.
- bluedanieru 14y ago
  Can you expand on this? I'm curious. It seems to me that with the glowworms, knowing the position of one worm tells you something about the position of every other worm in the sample. If it was truly random, wouldn't it tell you nothing?
  - tome 14y ago
    No, in mathematics the eye colour and hair colour of a randomly selected human being are considered random, even though knowing one tells you something about what the other is likely to be. Mathematically, "random" means distributed according to some definite (though perhaps unknown) distribution whereas in common parlance it means something like "not distributed according to anything definite at all" (though it's not clear that this way of defining things is even meaningful).
    - jwmerrill 14y ago
      Common parlance "random" seems to mean something more like "uniformly distributed and independent."
Xcelerate 14y ago
Here's a weird thing about randomness that has always bothered me: In quantum mechanics, if you measure two incompatible observables (like position and momentum) of a system, and then repeat that experiment many times, you will get two lists of real numbers. QM says you can predict the distribution of these numbers, but you cannot predict the individual numbers themselves. The popular way of thinking nowadays is that "the universe is just inherently random". So I posed the question on the Physics Stack Exchange: how do we know these numbers are truly random, and not the result of some as-yet-undiscovered pseudorandom number generator that is nonetheless deterministic? Luboš Motl (Czech string theorist) replied (a bit abrasively I might add) that yes, the numbers are truly random and plenty of experiments have ruled out the loopholes. Now, there's no way to determine if a set of numbers are truly random, so how he made this bold matter-of-fact statement is beyond me. Einstein initially believed in "hidden variable" theories, undiscovered properties of quantum systems. Most of these have been ruled out by experiment (this is what Lubos mentioned), but really, this doesn't apply at all to my question of whether those numbers are random or not. Superdeterminism seems to still allow non-randomness, but for some reason, most physicists (notably excepting Gerard t'Hooft) have discounted superdeterminism as nonsense.
- kragen 14y ago
  Motl, abrasive? Surprised? Maybe the issue is the Einstein-Podolsky-Rosen problem: if the numbers are being generated deterministically, they're somehow being communicated superluminally between entangled particles, which implies that in some relativistic frames of reference they're being communicated into the past? I guess I should learn enough about QM to really understand this stuff instead of guessing.
- pfortuny 14y ago
  Yes, that is the thing: Bell's inequality has been verified experimentally and that leads to a non-hidden variables reality (for the usual meaning of 'reality'): http://en.wikipedia.org/wiki/Bell%27s_theorem http://en.wikipedia.org/wiki/Bell%27s_theorem
  - sp332 14y ago
praptak 14y ago
You can try it out yourself in your browser. This scriptlet generates random and non-random dot distributions side by side (refreshing regenerates, at least in Firefox 17): javascript:"<html><body><canvas id=\"tutorial\" width=\"200\" height=\"200\">foo</canvas><-><canvas id=\"tutorial2\" width=\"200\" height=\"200\">foo</canvas><script>var canvas = document.getElementById('tutorial');var ctx = canvas.getContext('2d');ctx.fillStyle = \"rgb(000,0,0)\";for (var i=0;i<400;i++) {ctx.fillRect (Math.random() * 200,Math.random() * 200, 2, 2); };</script><script>var canvas = document.getElementById('tutorial2');var ctx = canvas.getContext('2d');ctx.fillStyle = \"rgb(000,0,0)\";for (var i=0;i<20;i++) for(var j=0;j<20;j++) for(k=0;k<1;k++) {ctx.fillRect (i * 10 + Math.random() * 10, j * 10 + Math.random() * 10, 2, 2); };</script></body></html>" (just paste the above into the address bar)
- roryokane 14y ago
  Here’s a web page displaying those dot distributions plus a more readable version of that code: http://bl.ocks.org/4358325 http://bl.ocks.org/4358325.
  - praptak 14y ago
    Thanks, this looks much better. My code was like this because I used a pasted canvas tutorial example + my browser's URL bar as the IDE :)
Avshalom 14y ago
As a side note this is one of those things that several tetris-clones deal have to deal with, long runs are probable but can frustrate a player so a lot of the times you want to to extra logic to keep it from being actually random. A uniform "deck" a couple times larger than the number of pieces is the usual suggestion prevents large runs and makes sure that you see every piece more regularly.
- wfn 14y ago
  Marginally related -- on the other hand, here's "Evil Tetris": http://qntm.org/hatetris http://qntm.org/hatetris
dgallagher 14y ago
Python script I wrote a while back that outputs random images. PIL must be installed to work. Change |file_name| accordingly. JPEG images can be up to 65535 x 65535 in size, which are the max values for |width| and |height|. It's not optimized, so keep the resolution small unless you want to wait a while. It'll output images that look like this: http://dave-gallagher.net/pics/666x666.png http://dave-gallagher.net/pics/666x666.png from PIL import Image, ImageDraw from random import randint def random_image(): width = 666 height = 666 file_name = '/Users/Dave/%dx%d' % (width, height) path_png = file_name + '.png' path_jpg = file_name + '.jpg' path_bmp = file_name + '.bmp' path_tif = file_name + '.tif' img = Image.new("RGB", (width, height), "#FFFFFF") draw = ImageDraw.Draw(img) for height_pixel in range(height): if height_pixel % 100 is 0: print height_pixel for width_pixel in range(width): r = randint(0, 255) g = randint(0, 255) b = randint(0, 255) dr = (randint(0, 255) - r) / 300.0 dg = (randint(0, 255) - g) / 300.0 db = (randint(0, 255) - b) / 300.0 r = r + dr g = g + dg b = b + db draw.line((width_pixel, height_pixel, width_pixel, height_pixel), fill=(int(r), int(g), int(b))) img.save(fp=path_png, format="PNG") img.save(fp=path_jpg, format="JPEG", quality=95, subsampling=0) # 100 quality is 2x to 3x file size, but you won't see a difference visually. img.save(fp=path_bmp, format="BMP") img.save(fp=path_tif, format="TIFF") if __name__ == "__main__": random_image()
ChrisNorstrom 14y ago
True Randomness = If you roll a six sided die 60 times, each side will NOT get chosen 10 times. 1 side(s) will get chosen more than others, another side(s) might not get chosen at all. So for all you entrepreneurs, if you fail, don't fall into a depression. Sure you worked just as hard as everyone else, maybe even harder. And yeah it's annoying to see others surpass you even though you've got everything they do. But that's life, you just got a bad batch of rolls.
jebblue 14y ago
I was happy that the first example with the dots looked like the second one was natural. My eyes felt better. With the student example my eyes struggled so I "reasoned" that the first one had more clumps and was wrong since the first dot example had more clumps and was wrong. I guess I'm not a good randomness detective.
ashutosh2000 14y ago
This article is really interesting. It says that randomness doesn't mean uniformity but true randomness can have clusters. So what exactly is randomness? How can we define randomness? Is Math.random() is really random? Which is the best random function and how can we find if it is purely random?

474 points by cyrusradfar 14y ago

thaumaturgy 14y ago

I would love to see more like this on HN.

cyrusradfar 14y ago
The author is @aatishb (https://twitter.com/aatishb https://twitter.com/aatishb) on Twitter. I've followed him for some time and he's a enjoyable science/math blogger if you're interested in that.
- thaumaturgy 14y ago
  Thank you! I've added him to my reading list.

suby 14y ago

"The one on the left, with the clumps, strands, voids, and filaments is the array that was plotted at random, like stars." Are stars really plotted at random?

davidcuddeback 14y ago
That's a good question. There's obviously more stars in the directions of the galactic plane (what we observe as the Milky Way), but the individual stars that we see in the night sky tend to only be the ones that are really close to us (relatively speaking). So it's a question of which effect dominates the other. I think it'd be interesting to see how well the distribution of stars fits a Poisson distribution.
- morsch 14y ago
  Imagine you were one of the dots on the plane in the dot illustrations. Would the "constellation" of all the other dots from your position be Poisson distributed?
pixl97 14y ago
http://en.wikipedia.org/wiki/Observable_universe#End_of_Greatness http://en.wikipedia.org/wiki/Observable_universe#End_of_Grea... Up to the largest scales the universe is randomly clumped/stringed/voided.
Devilboy 14y ago
Well no, if you look at the whole sky then obviously you'd see more stars in the direction of the galactic centre (the milky way) than in the opposite direction, but if you consider only a small sliver of sky the results would be pretty close to being truly random.

gluegeorge 14y ago

Were the bombs actually mostly random?

deleted 14y ago
[deleted]
rosser 14y ago
Without asking the folks aiming them we can never be sure, but the results of the survey of South London detonations mentioned in the article was shockingly close to the Poisson distribution. So it seems likely.
davidvaughan 14y ago
No, they were aimed but it was a matter of trial and error. To get the data about where they'd landed, some had radio transmitters. They sent a message back to armourers in France (I assume some were deliberately non-explosive). Meanwhile German agents in England were also observing the success or failure of the V2s, and in particular where they'd landed. However, in an effort to deceive the Germans, the British started reporting the correct time of successful attacks, while mentioning an incorrect location. Moreover, a double agent called Eddie Chapman also fed false information back to the Germans. As a result, the aimers never really got a grip on ranging accurately. The bombs started landing to the south east of London. There's considerably more detail about this in Most Secret War by R V Jones, who was involved in all sorts of ruses to confuse the enemy. Well worth a read. Eddie Chapman (Agent ZigZag) was played by Christopher Plummer in the film Triple Cross. It seems to be on YouTube.
- andrewcooke 14y ago
  does that book discuss whether / how the misinformation was used to select the south east? i suspect that was a poorer area of london and i vaguely remember some kind of scandal about the relative suffering of various parts of london and class, etc. so i wonder to what extent the south east was chosen (by the british) as a target?
InclinedPlane 14y ago

pm90 14y ago

Very interesting! So, the basic point being made is that if you know that a set of events are random and independent and you know their mean value, then you can predict their spread? (or aggregation) edit: Hmm...another question that comes to mind: is the converse true? If the spread of values of these events do not match the poisson distribution, then can we presume them to be nonrandom? Or nonindependent? Or both?

quantumet 14y ago
The Poisson distribution is just one random number distribution; there are several others for situations where the events are correlated, or have other properties. Half the fun of probability is figuring out which distribution is the right one to apply to the question at hand. So if your measurements don't match up to Poisson, it doesn't mean they're not random - they could just be interdependent. So yes, for a Poisson process, the spread (standard deviation) is equal to the square root of the mean; as the number of events gets large, the Poisson distribution approaches the normal distribution, but the relationship between the standard deviation and the mean continues to hold.
Persephone404 14y ago
If you have enough data and to determine the distribution and it's characteristics (eg mean and s.d. for normal etc) then you can use it to predict spread.

tdyo 14y ago

This is exactly how they determine species distributions in ecology as well - http://en.wikipedia.org/wiki/Species_distribution http://en.wikipedia.org/wiki/Species_distribution Uniform dispersion would suggest some territoriality aspect of the species, and clumped dispersion would suggest a heterogeneity of resources (or any other hypothesis that could then be tested).

aatish 14y ago
Nice! That's a really interesting and relevant example.

henrikschroder 14y ago

Back in the day I was in a raiding guild in World of Warcraft. And back then, each boss monster in a raid dungeon had a loot table, and when you killed it, your raid got a few pieces of random loot. And I don't know how many times I, and the other guy in the guild who understood statistics, had to explain to the others that random doesn't mean uniform. If a certain piece of armour has a 1/X chance to drop from a boss, what people think should happen is that if they kill that boss X times, they should see it drop once. But the reality was of course that loot was very non-uniform. Some pieces we saw lots of times, and other pieces very rarely, despite them having the same drop chance. And the players who wanted those pieces that happened to be rare for our guild, got very, very angry. We saw the same things on the official message boards, players were furious after having spent a year killing the same raid boss once a week, and never seeing a certain piece drop for them. But simple math shows that with million of players, tens of thousands of raiding guilds, some of those will see very streaky results. These days in World of Warcraft, boss monsters drop tokens instead, and when you have X tokens, you can exchange that for a piece of armour, or a weapon, guaranteed. And noone complains about the random loot anymore.

thurn 14y ago
This is actually an important lesson for game designers. Real randomness is very frustrating for players! Games should be designed to not be random. For example, there's an expansion to The Settlers of Catan that lets you use cards instead of dice to ensure a nice smooth distribution of resources... I think it's a lot more fun!
- henrikschroder 14y ago
  Oh yes. Our intuition is that random is fair and uniform, but it's absolutely not. I was bit by another random quirk in World of Warcraft. There was a long-running achievement called "What a Strange Long Journey it has Been", which took at least a year in real-time to complete, and you had to actively play during each of the ten or so in-game festivals and holidays, and do some quests and tasks during each. If you missed a festival, you had to wait another year to do it again, so if you were aiming for the achievement, you really wanted to do it in one go. During their version of Valentine's, each player got a bag of heart-shaped candy, and the task you had to complete was to pull out at least one each of the eight different heart candies. But you could only pull a piece of candy once every hour or so, each time you pulled a piece you had a 1/8 chance to get a certain piece, but the holiday was time-limited, so you had about two weeks to complete it. Sounds easy and fair, right? 1/8 chance, get all eight pieces, two whole weeks, easy! Except that there was one piece that I just never got. The piece that said "I LOVE YOU!". And as the time went by, I got more and more frantic, logged in more often so as not to miss any opportunity to pull a piece of candy, but no luck. So, I did a quick bit of math. You can pull one piece every hour for two weeks, but sleeping, not playing, missing days etc, meant that I effectively pulled ~100 pieces. The chance of missing a certain piece is 7/8, so missing a certain piece 100 times in a row is (7/8)^100, which is roughly a little bit more than one in a million. With ten million players all doing the same thing, there's going to be a number of them that will hit that "one in a million" chance, which means that whoever designed that part of the achievement didn't do their homework, didn't do the math. Because intuition tells you that 1/8 chance is plenty! Hundreds of tries, of course everyone will get all pieces! Except proper math tells you otherwise, and I was one of the "lucky" outliers. (Later, they apologized and retroactively removed that part of the achievement, so I got my purple nether-drake mount without having to wait a year extra.)

marvin 14y ago

Regarding the bomb plot at the very bottom. Examining the number of bombs dropped in each square and comparing it to the Poisson distribution, it appears that the distribution of the bombs is random. But looking at the plot on the map, it appears that the higher incidence of bombs is focused on a specific region. The Poisson distribution doesn't account for the fact that a lot of the squares with a high incidence of bombs are adjacent to each other. From my layman's understanding, it appears that the bombs were in fact targeted on a specific area, but that there was a random offset from this area regarding where the bombs actually landed. Because of this, you'd see randomness in the distribution. But the distribution of bombs wasn't really perfectly random. Is the author deliberately avoiding this point, or is there something I've misunderstood?

patrickod 14y ago
I believe the table at the end deals with a subsection of the 3d graph, but I could be wrong.
feral 14y ago
If you read the linked post its clearer whats going on: http://madvis.blogspot.ie/2010/09/flying-bombs-on-london-summer-of-1944.html http://madvis.blogspot.ie/2010/09/flying-bombs-on-london-sum... The poisson analysis is only done on a subset of the area thats in the 3d plots: "Clarke's analysis was focused on the central area of higher density here and with finer geographic coordinates. Within that area his analysis found no evidence of clustering that cannot be accounted for by a Poisson process."
- jerf 14y ago
  Yes, it could have been a bit more clear in how they "weren't being aimed". Obviously the Germans did not just point them straight up and light them off, hoping for the best. What this really demonstrated was something more like where the noise floor for the bomb distribution was. They could hit "London" but they couldn't hit any particular place in it. Very important for the military to know that information.
aatish 14y ago

y4m4 14y ago

Such a great article! - something nice at the HN top after a long time.

saosebastiao 14y ago

TL;DR http://www.google.com/finance http://www.google.com/finance

Zenst 14y ago

True randomness is a system that say generates a 10 digit number and 11111111 has the same odd's as appearing as the other permutations. But then thats not strictly true as we all know any system that monitors the outcome (a good example will be fruit machines) to make sure there is a even distribution of all permutation will in essence remove that level of definition we like to think as truely random. With that it gets hard to truely say what is random or what is a as yet unknown pattern. This is why many have taken the approach of not having a single source of random numbers but use many and average out from there. There again is that random as the chances with such an approach of getting a high or low value would be biased out. So with that I postulate one mans random string is another mans non-random string. So with that I define randomness as a yet undertermind sequence or a data. So the included Dilbert post is with that extreemly clever and totaly true.

Steko 14y ago
"hard to truely say what is random or what is a as yet unknown pattern." No, it's just saying that it's much easier to get pseudorandomness out of a computer than true randomness. "This is why many have taken the approach of not having a single source of random numbers but use many and average out from there. There again is that random as the chances with such an approach of getting a high or low value would be biased out." Technically speaking you wouldn't average you'd add them together and take the decimal portion (modulo 1). That can negate bias as long as one of the sources is good even if you don't know which one. Of course you can remove bias from a single source by Von Neumann's method although this might be computationally harder than the above: http://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_biased_coin http://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_b...
Dylan16807 14y ago
There is actually no such thing as a random or not random number. There are only random and not random generators. Whether a fruit machine is random depends on how is defines/ensures evenness.

laureny 14y ago

Here is an article on how to assess whether your data sample is "random enough": http://goo.gl/opl52 http://goo.gl/opl52

michaelochurch 14y ago

One under-stated but awesome thing about Poisson distributions: fishing has a Poisson distribution. (Poisson is French for "fish".)

endtime 14y ago
I think that would only be true in an infinite lake.

Uhhrrr 14y ago

"Gravity's Rainbow", by Thomas Pynchon, has an extended section about the Poisson distribution as applied to falling bombs, maternity wards, etc.: http://books.google.com/books?id=GGPm4I3BbxAC&pg=PT194&lpg=PT194&dq=poisson+distribution+gravitys+rainbow&source=bl&ots=APsLuTKC1F&sig=wUaE1csdIbr744iR-_ZvRVQ1sCo&hl=en&sa=X&ei=KBfVULfTI8OFiAKT04CgDw&ved=0CGIQ6AEwBQ http://books.google.com/books?id=GGPm4I3BbxAC&pg=PT194&#... (possibly NSFW)

krymise 14y ago
That book is exactly what I thought of when reading the article. Everyday I learn something new that seems to unlock a section of that novel that was previously opaque.

baddox 14y ago

I'm a bit confused. What is this notion of "pure randomness" that this article and many comments seem to be eluding to? Perhaps I'm just not using the same definitions of terms, but I thought a uniform distribution is still considered random.

lutusp 14y ago
> Perhaps I'm just not using the same definitions of terms, but I thought a uniform distribution is still considered random. No, a uniform distribution is not evidence of randomness. Consider the digits 0 - 9 repeated endlessly: 0123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789 ... Uniformly distributed? Yes. Random? No. > What is this notion of "pure randomness" that this article and many comments seem to be eluding [sic] to? First, s/eluding/alluding/ Second, although the topic is complex, one test of randomness is that an ideal compression method, one able to find and exploit any repetitive pattern, cannot compress a random sequence. Third, the term "entropy" as used in information theory is tied to randomness, as explained here: http://en.wikipedia.org/wiki/Entropy_(information_theory) http://en.wikipedia.org/wiki/Entropy_(information_theory) A quote: "The entropy rate for a [fair coin] toss is one bit per toss. However, if the coin is not fair, then the uncertainty, and hence the entropy rate, is lower." Based on that, high entropy -> high randomness. Not to oversimplify a complex topic.
- pronoiac 14y ago
  Could you edit that long line somehow? It makes reading harder in my phone.
- endtime 14y ago
  >Uniformly distributed? Yes. Random? No. I think you're fudging what is supposed to be uniform here. In your example, the unigrams (i.e. single digits) may be uniform, but the n-grams for n > 2 are not.
  - lutusp 14y ago
    Actually, the n-grams are periodic and repetitive also. They certainly aren't random.

tome 14y ago

Strange terminology. In mathematical parlance the arrangement of the glowworms is still described as "random", but the positions of the worms simply are not independent of one another.

bluedanieru 14y ago
Can you expand on this? I'm curious. It seems to me that with the glowworms, knowing the position of one worm tells you something about the position of every other worm in the sample. If it was truly random, wouldn't it tell you nothing?
- tome 14y ago
  No, in mathematics the eye colour and hair colour of a randomly selected human being are considered random, even though knowing one tells you something about what the other is likely to be. Mathematically, "random" means distributed according to some definite (though perhaps unknown) distribution whereas in common parlance it means something like "not distributed according to anything definite at all" (though it's not clear that this way of defining things is even meaningful).
  - jwmerrill 14y ago
    Common parlance "random" seems to mean something more like "uniformly distributed and independent."

Xcelerate 14y ago

Here's a weird thing about randomness that has always bothered me: In quantum mechanics, if you measure two incompatible observables (like position and momentum) of a system, and then repeat that experiment many times, you will get two lists of real numbers. QM says you can predict the distribution of these numbers, but you cannot predict the individual numbers themselves. The popular way of thinking nowadays is that "the universe is just inherently random". So I posed the question on the Physics Stack Exchange: how do we know these numbers are truly random, and not the result of some as-yet-undiscovered pseudorandom number generator that is nonetheless deterministic? Luboš Motl (Czech string theorist) replied (a bit abrasively I might add) that yes, the numbers are truly random and plenty of experiments have ruled out the loopholes. Now, there's no way to determine if a set of numbers are truly random, so how he made this bold matter-of-fact statement is beyond me. Einstein initially believed in "hidden variable" theories, undiscovered properties of quantum systems. Most of these have been ruled out by experiment (this is what Lubos mentioned), but really, this doesn't apply at all to my question of whether those numbers are random or not. Superdeterminism seems to still allow non-randomness, but for some reason, most physicists (notably excepting Gerard t'Hooft) have discounted superdeterminism as nonsense.

kragen 14y ago
Motl, abrasive? Surprised? Maybe the issue is the Einstein-Podolsky-Rosen problem: if the numbers are being generated deterministically, they're somehow being communicated superluminally between entangled particles, which implies that in some relativistic frames of reference they're being communicated into the past? I guess I should learn enough about QM to really understand this stuff instead of guessing.
pfortuny 14y ago
Yes, that is the thing: Bell's inequality has been verified experimentally and that leads to a non-hidden variables reality (for the usual meaning of 'reality'): http://en.wikipedia.org/wiki/Bell%27s_theorem http://en.wikipedia.org/wiki/Bell%27s_theorem
- sp332 14y ago

praptak 14y ago

You can try it out yourself in your browser. This scriptlet generates random and non-random dot distributions side by side (refreshing regenerates, at least in Firefox 17): javascript:"<html><body><canvas id=\"tutorial\" width=\"200\" height=\"200\">foo</canvas><-><canvas id=\"tutorial2\" width=\"200\" height=\"200\">foo</canvas><script>var canvas = document.getElementById('tutorial');var ctx = canvas.getContext('2d');ctx.fillStyle = \"rgb(000,0,0)\";for (var i=0;i<400;i++) {ctx.fillRect (Math.random() * 200,Math.random() * 200, 2, 2); };</script><script>var canvas = document.getElementById('tutorial2');var ctx = canvas.getContext('2d');ctx.fillStyle = \"rgb(000,0,0)\";for (var i=0;i<20;i++) for(var j=0;j<20;j++) for(k=0;k<1;k++) {ctx.fillRect (i * 10 + Math.random() * 10, j * 10 + Math.random() * 10, 2, 2); };</script></body></html>" (just paste the above into the address bar)

roryokane 14y ago
Here’s a web page displaying those dot distributions plus a more readable version of that code: http://bl.ocks.org/4358325 http://bl.ocks.org/4358325.
- praptak 14y ago
  Thanks, this looks much better. My code was like this because I used a pasted canvas tutorial example + my browser's URL bar as the IDE :)

Avshalom 14y ago

As a side note this is one of those things that several tetris-clones deal have to deal with, long runs are probable but can frustrate a player so a lot of the times you want to to extra logic to keep it from being actually random. A uniform "deck" a couple times larger than the number of pieces is the usual suggestion prevents large runs and makes sure that you see every piece more regularly.

wfn 14y ago
Marginally related -- on the other hand, here's "Evil Tetris": http://qntm.org/hatetris http://qntm.org/hatetris

dgallagher 14y ago

Python script I wrote a while back that outputs random images. PIL must be installed to work. Change |file_name| accordingly. JPEG images can be up to 65535 x 65535 in size, which are the max values for |width| and |height|. It's not optimized, so keep the resolution small unless you want to wait a while. It'll output images that look like this: http://dave-gallagher.net/pics/666x666.png http://dave-gallagher.net/pics/666x666.png from PIL import Image, ImageDraw from random import randint def random_image(): width = 666 height = 666 file_name = '/Users/Dave/%dx%d' % (width, height) path_png = file_name + '.png' path_jpg = file_name + '.jpg' path_bmp = file_name + '.bmp' path_tif = file_name + '.tif' img = Image.new("RGB", (width, height), "#FFFFFF") draw = ImageDraw.Draw(img) for height_pixel in range(height): if height_pixel % 100 is 0: print height_pixel for width_pixel in range(width): r = randint(0, 255) g = randint(0, 255) b = randint(0, 255) dr = (randint(0, 255) - r) / 300.0 dg = (randint(0, 255) - g) / 300.0 db = (randint(0, 255) - b) / 300.0 r = r + dr g = g + dg b = b + db draw.line((width_pixel, height_pixel, width_pixel, height_pixel), fill=(int(r), int(g), int(b))) img.save(fp=path_png, format="PNG") img.save(fp=path_jpg, format="JPEG", quality=95, subsampling=0) # 100 quality is 2x to 3x file size, but you won't see a difference visually. img.save(fp=path_bmp, format="BMP") img.save(fp=path_tif, format="TIFF") if __name__ == "__main__": random_image()

ChrisNorstrom 14y ago

True Randomness = If you roll a six sided die 60 times, each side will NOT get chosen 10 times. 1 side(s) will get chosen more than others, another side(s) might not get chosen at all. So for all you entrepreneurs, if you fail, don't fall into a depression. Sure you worked just as hard as everyone else, maybe even harder. And yeah it's annoying to see others surpass you even though you've got everything they do. But that's life, you just got a bad batch of rolls.

jebblue 14y ago

I was happy that the first example with the dots looked like the second one was natural. My eyes felt better. With the student example my eyes struggled so I "reasoned" that the first one had more clumps and was wrong since the first dot example had more clumps and was wrong. I guess I'm not a good randomness detective.

ashutosh2000 14y ago

This article is really interesting. It says that randomness doesn't mean uniformity but true randomness can have clusters. So what exactly is randomness? How can we define randomness? Is Math.random() is really random? Which is the best random function and how can we find if it is purely random?