I know of a teacher once who instructed her students to go home and do one of the following:
a) Flip a coin 100 times and record the sequence of heads and tails
b) Record a sequence of heads and tails "randomly" that is supposed to simulate coin flips

When they came back the next day with their results, she was able to tell with remarkable accuracy whether they had actually flipped the coins or just pretended to flip the coins. Human beings are lousy at trying to make numbers seem random. The actual coin flip sequences would have some streaks of several heads or several tails in a row, while humans tended to avoid the longer such streaks. Having 4 heads in a row seems wrong to many people. Obviously the probability of a flip being different from the preceding flip is 0.5, but I believe I read somewhere that humans tend to make this about 0.7 when faking it.

Another example (used on the show NUMB3RS), is that if people are told to spread out "randomly," they actually tend to space themselves at roughly equal distances. It seems that people have this intuitive idea that being "random" means that "clusters" should be avoided. Humans seem to have an expectation that clusters or streaks are marks of nonrandomness, even though such streaks are to be expected some portion of the time.

Does anyone have a good idea why people tend to do this? I think that poker players and those good at math are less likely to succumb to this thinking, but there are certainly going to be exceptions (especially considering one of the all-time greats believes in the related notion of "rushes" in poker).

I think because humans tend to have a lousy understanding of random numbers, we often tend to look for explanations for things that might be better explained by chance. For instance, after a particularly good winning session it might be easy to convince ourselves of how we really played our "A-game" to focus on opponents and make good decisions, and although this might be the explanation some of the time, much of the time it is also due to the cards falling our way a little better than average that time.

i think the problem, at least in the large group example, is that everyone thinks the same. if you have a few people thinking of grouping and a few not then it will work, but in groups people tend to think similarly. with the individuals humans have trouble believing random things are actually random, so i doubt they would be much better at generating something they can't recognize in the first place. i think on the intuitive practical mathematics spectrum humans are the best at geometry and worst at probability. we just have no reason to be tuned for the idea of randomness as most of what we encounter isn't very random and has a pattern to it. with geometry having an understanding of how to maximize areas and what shapes work best for certain things both a) comes up in nature a lot and b) is practically useful to humans.

I understand where you are coming from with the geometry-probability comparison. However, it seems to me that we aren't just talking about people lacking an ability to understand randomness, but it seems like we have adapted a sort of "negative ability" for it. It's not just that half the people make sequences with too many streaks and half with not enough streaks, but consistently we tend to be "nonrandom" in the same manner. It seems like this is some sort of "ability" that we acquired at some point, though I'm not sure why there'd be incentive to become bad at something like this.

Humans who are told to generate a random series of numbers will tend to avoid putting equal numbers after eachother. What they often try to do is to avoid making a pattern (pattern here means something human perception sees as a pattern), but in a truly random sequence of number the probability is very high there will also be sequences that looks like patterns.

So a human generated strings of "random" numbers will have fewer equal numbers after eachother than what is likely (I don't know the terminology of how to say it).

You can try this with friends and family by telling them to put 20 random numbers in a row or some such.

that is just what i am saying, we are bad at it because we have evolved to look for patterns. hell, this was in the first chapter of SSNL by sklansky, we are results oriented. we have evolved to be really good at seeing patterns. in my mind it makes perfect sense that we suck at randomly generating numbers because we think in patterns, our brain is full of little shortcuts we take by using patterns. if you had to take in all the information of your surroundings to operate your brain would be full within a day, so we assume the vast majority of the stuff in our world follows some familiar pattern. the problem comes when there really isn't a pattern, our brain can't detect this and looks for patterns that don't exist.

Is this not the same thing I said in the OP, down to the fact that the teacher played the game with students (albeit 100 random coin flips instead of 20 random numbers)?

I guess it is.

If you want to know why, then it is because the human brain is geared towards pattern recognition.

Pattern recognition is a cheap and effective (it conserves processing power, energy and time) way of handling large amounts of data and will most likely give an evolutionary advantage.

"Snakes = bad" is simply better than treating each snake individually. You can see other the side-effects clearly enough in effects such as the "gambler's fallacy" - the human brain tends to see patterns also where there are none.

Still all things considered the pros outweigh the cons and we can imagine in an natural setting then an organism which gains a neural network that uses pattern recognition is superior to one that doesn't.

An analogy using computers would be data compression, fitting similar patterns together has advantages and disadvantages. (It is not a perfect analogy but it works)

obviously it's impossible for a human being to come up with a truly random number sequence (on his own). i don't know why it's surprising that imitation-random sequences by uneducated people are distinguishable from random sequences.

but yes, it is kind of amusing

Humans are good at finding and creating patterns. So when humans try to create randomness they do so following a pattern/method. Randomness by definition does not follow patterns.

I once questioned students to randomly create sequences for the game mastermind, and we were left with 2 extremes. One smaller group who thought it was funny to create sequences like 1111 1111 11111 11etc. And another group that almost never repeated numbers right next to eachother. They almost felt it disobeyed the randomness phenomena.

Another peculiar outcome from our questionaire was that when people choose randomly between 1 and 4, over 80% pick number 3.

Checking if a string is human generated or random generated can be as easy as checking the complexity of the string.

Right, it's the pattern-generating thing.

I think a good way to look at it is from the opposite perspective. If you show an untrained person different strings of numbers, I'd bet the strings he considers most likely to be random are actually the ones where the bias has been applied! I'd bet an untrained person (rather than a probability instructor) would tend to pick out the "streaky" sequences as the ones that "aren't really random" and have a strong sense of confidence in their selections (despite being horribly wrong). So in a sense, we choose less random sequences because we understand that they are usually more likely to be perceived as random (ironically, the cheating students probably thought "if I include a streak of 4 heads in a row, the teacher will know I'm cheating").

I'm taking a math class right now that covers basic probability, and when our teacher was going over frequency distributions he ran a computer program that rolled dice randomly and displayed a histogram with frequency.

The first time he ran it the program rolled 6 many times in a row, and of all the (I think it was 10) rolls, only 6 and 5 were ever rolled. He showed that even when rolling 4 dice (4-24) 1000 times, the results were heavily skewed on a regular basis. Also, he ran a 10,000 coin flip program and got a frequency of 50.2 or something, and then ran a 100,000 coin flip program and got a frequency of over 51. Apparently when he runs these various demonstrations, he almost always gets freaky results from at least a couple of the simulations.

I think a lot of students really had their eyes opened, and I think some were suspicious of whether the random number generation was accurate. I mean, even I (who knew this stuff already) had to fight a natural impulse to think there was a problem with the number generation. When I was young, I also used to frequently think my computer games had poor rngs, because I got streaks much more often than I should have. It wasn't until later that I realized I had no basis for having a sense of what results I "should" have had. Humans just don't like to admit that seeing a pattern doesn't always imply a pattern. Just think about the people who come into 2+2 and ask "what is the chance that a guy with T4 would get a 444 flop," or the perception of cheating and rigged games that is so ubiquitous

On another note, I know when asked to choose a number from 1 through 10, people are much more likely to choose 7 than any other number. Anyone have insight into that?

Oh, same deal. What's that all about?

It makes sense for the same reasons. The first and last numbers of a distribution don't appear "random" to a normal person, and the second number suffers from the same problem to a lesser extent (no one is going to pick "2" when choosing from 1-100). "3" is logically and aesthetically the most random number of the four.

7 is just known as a lucky number.

3 and 7 are common cultural numbers. 7 days of the week, 3 holy men, trinity, 3 stooges, "lucky number seven", 7 samurai, think of all the jokes you know who has 3 characters, "third time is the charm"....well the list goes on and on.

For encoding in memory 3 bunks (pieces) of information is also alot easier to remember than 4, but not much more difficult than 2. More than 7 bunks of informations means information is likely to be lost when coding to long term memory (an unprecise expression, but it is decent enough for explaining). I don't know if that has anything to do with it - but I guess it could explain the prevalance of these numbers in stories.

But 3 is to 4 what 2 is to 1.

Conversely, humans are also good at being deceived into thinking that something which is actually random contains patterns, order and meaning.

Yes, i think it partly have to do with some kind of association. If a random person thinks of a number between 1 and 10, the first number that pop into his mind is probably 3 or 7 (cultural reasons stated above?). Its like the game where you ask someone to think of a colour and a tool really fast and most people end up with a red hammer. But if you insists that the person take 1 min before answering (giving him time to think of other things then the first thing that comes to mind), I bet there would be a much greater variation in the answers (but far from random in the 1-10 case).

Another reason might be that the number 7 seems random in a 1-10 interval. Its odd and its not in the middle or on the ends of the interval. It probably makes sense to our pattern seeking brain (specially if you dont know much probability and know why that logic is flawed).

My prof gave it not the cultural number preference-explanation, but that the question itself is more or less biased and suggestive.

Even with open questions such as 'Pick One', those who answer tend to want to give the 'right' answer. Out of the sequence '1 2 3 4', 1 and 4 tend to be crossed away as the outliers. (Pick a number between 1 and 4).

2 is too transparant, and 3 just feels right. We are just very awful at picking 'truelly' random, else we would see a normal distribution.

If I remember correctly it was about:
4 1%
1 2%
2 15%
3 82%

Chance and randomness don't come naturally to most people. Some people believe rolling a die 3 times and getting the sequence '666', rule the sequence '666' to be less likely than any other 'less patternized' sequences.

It is not shocking that these same people are more likely to be religious. I for one am tempted to play Tarot with the universe and my surroundings.