I will start to teach an introductory statistical physics in a few weeks. One of the things I will do is talk about why in a still room the air is uniformly distributed in the room. The reason is statistics, to take an extreme example there is no law stopping all the gas molecules in a room being in the left-hand half of the room (gases are dilute so there is plenty of room to pack all the molecules in only one half of the room). It is just that the probability of this occurring is approximately (1/2) raised to the power of the number of molecules in the room.
This is an unimaginably small number, a probability that is so ridiculously small that it is utterly unlikely to occur anywhere in the universe, even if the universe lasts a trillion years. It is a nice demonstration of the power of stats, at least I hope it is.
Anyway, it involves an approximation: the assumption that the molecules are independent of each other. This is an OK approximation in a gas like the air where the molecules are mostly far apart, so although this approximation will introduce an error into this approximation, the error is not serious.
The probability is still so low that you or I will never live to see it happen.
However, people who have applied statistical models to the finance have not been as careful. Of the 2008 crash, David Viniar, CEO of Goldman Sachs, said: “We were seeing things that were 25-standard deviation moves, several days in a row.” [Note for those familiar with stats: yes that really is a 25.]
Here the standard deviation is a measure of the size of the fluctuation from the typical value. In the example of a gas, the typical value of the fraction of molecules in the left-hand half of the room is a half, and a fluctuation is any move away from this value of a half.
Now, the standard model, that I used for the gas, predicts that the probability that you find a value that is one standard deviation larger than the typical value is about 1 in 6, or 16%. So, deviations of 1 standard deviation are common.
However, deviations of more than 3 standard deviations only occur once in 30 times. They are much less common.
Deviations of 25 standard deviations should be seriously rare. If you look once a day, then deviations of 25 standard deviations occur once in every approximately 1 followed by 135 zeros years. The age of the universe only has 10 zeros in it, so this is a seriously long time.
It is such a large number it broke the calculator application on my Mac. Note Goldman Sachs CEO saw such deviations several times. His model would predict that that should not happen in trillions of lifetimes of the universe.
When an event that should not occur in the lifetime of the universe, occurs several times in the same week, this is pretty clear evidence that the model Goldman Sachs were using was wrong. Not just a little bit wrong but wrong by an absolutely huge amount. The reason is simple: correlations. Just as I assumed that the molecules are independent (a good approximation), Goldman Sachs assumed that the probability of people defaulting on their mortgages were largely independent (a terrible approximation).