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Thursday, April 30, 2009

A common cognitive error

Patterns and Randomness

Lottery

Let us again look a bit at the lottery in the Norwegian Lotto. You have to choose seven numbers, out of 34. Some choose special numbers like birthdays, anniversaries, ages, their house number and so on. Other choose random numbers. Or at least seemingly random, as our brains are notoriously bad at doing random.


Here is the numbers from the last 52 drawings of Lotto, 364 in all. We can see that there is quite the variance in the numbers frequencies. For instance 3 has been picked over twice as often as poor little 16! If you are going to play the lottery you surely should pick that one, as it's on a hot streak! Or... shouldn't we rather pick the ones that haven't been picked in a while, they must be due soon?

The correct answer is of course neither. Former results have no effect on what will be picked next. (Lotto takes special care that even things like wear and tear on the balls shall not affect the results)

Here's all the Lotto results from 1986 to today, 1196 drawings total.


Here we can see a slightly smoother distribution, but there are still peaks and nadirs. You will never see a totally flat distribution, not even with millions of iterations. Though the more we do the flatter it will get.


Coin flips

Ah, now we're talking. Nothing quite beats the coin flips for experiments with randomness.
Let us flip ten coins after one another, noting down the results as we go along. Of the following two results, which one is one that you are more likely to see?
(H=Heads, T=Tails)

A. HTHHTTHTHT
B. TTTTTTTTTT

Most of us (at least those of us without statistics education :) will answer A, because it looks "more random." In addition, it has an equal number of heads and tails, and given 50% chance for heads or tails we would expect approximately that.
But it would be dull if everything was as we thought it were! In fact, either result is exactly as probable to be seen in a random toss of ten coins. (Don't let yourself fool that ten tails in a row only has a 1/1024 chance, the exact sequence in A has also the same chance)
It is highly unintuitive for us to think about it this way. It will get even harder for us if we take the following exercise instead:

I will flip nine coins after one another, and record the flips. You try to guess what the tenth will be. Here is the nine flips in three different scenarios:

A. HHTHHTHHH
B. TTHTTTTTT
C. THTHTHTHT

And of course again in all three of these the right answer is 'It is 50% chance for either heads or tails in all three scenarios.' But for most of us it is hard to imagine that right at the top of our heads. We try to see a pattern. For C you would like to say that the next one is Heads, as they have alternated heads and tails so far. Set B, after you have examined that the coin is not weighted of course, you'd like to think that it's due for a Heads soon. A also had a pattern, but it is a less apparent one, so it is more likely that you will just pick randomly rather than try and 'guess' which it will be.

Choosing randomly
Exercise, and do this without too much deliberation:
Think of a random number between one and ten (inclusive). Do this now before reading on.





I will state with about 75% accuracy that you chose either 3 or 7. The reason you chose either one of those is because your brain wanted it to look like you chose something random, something which is really hard for your brain to do. In wanting to choose a number that "looks random", you mentally eliminated all the other numbers for "not looking random." (How does a single number look more or less random than another, anyway?)
1 and 10 are in the extremes of the range, so they do not "look random." 5 is in the middle and thus is not random either. And for some reason, the even numbers 2, 4, 6 and 8 seem less random than the odd ones. That leaves 3 and 7 as the only two "random" numbers between 1 and 10.1

Had our brain been able to think randomly then the numbers 3 and 7 would not have been greatly overrepresented whenever someone takes this "test." The number 1 is exactly as random as any other.

Check this one out. Click it a couple of times and see if you think that the numbers generated look "random." They probably don't. Your brain will try to see a pattern in it. A pattern that does not exist, because those numbers there definetly are truly random.

We're wired from nature to try and see patterns. It's one of the things that has helped us survive as a species so far. Unfortunately it is very prone to false positives, to see a pattern where there is none. We see patterns in random numbers. We think that lottery numbers can be predicted. We also are not able to do random things ourselves.

Watch out. When coming from humans, random things are usually not.



This is actually correct.


1Did you notice I left out nine? There is a reason for that but I won't spoil that yet.

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