Title: The distance between chance and structure
Course: UGQuants (SSPS) > Section: Probability

The distance between chance and structure

Given we know what 'shape' chance takes we can measure the 'distance' between the shape of various phenomena as against that of chance. The bigger the distance the stronger the evidence that we have encountered some kind of social regularity that merits our attention, rather than a purely random pattern analogous to the outcome of tossing a coin.

For example, if we discover a situation in which a coin has been tossed one hundred times and resulted in tails every time, it is possible that the coin is true, but extremely unlikely. We might want to investigate whether there was some structural phenomenon responsible for this highly improbable (but nevertheless theoretically possible) event: a corroded coin, outside manipulation etc.

How do we know whether what we have witnessed constitutes a relationship, pattern or regularity? How do we know that it is evidence of some kind of structure or order, rather than a pattern thrown up by pure chance?

The answer, unfortunately, is that we can never know absolutely. It is always theoretically possible to toss a 100 tails, or for a monkey to randomly type out the title of a Shakespeare play. However, if someone handed us a sheet of paper with the first few lines of King Lear typed on it, and asked us to guess if it had been produced by a monkey or Shakespeare (assuming, anachronistically that Shakespeare not only had a typewriter but passed on his manuscripts to people we know) we'd be on a fantastically sure bet if we said 'Shakespeare'.

Why? Because the probability of writing a few lines of anything coherent at all, let alone a specific set of words, by choosing letters at random is so incredibly small. The same holds for with any structure, order, relationship or regularity. We can define it as something so unlikely to occur by chance that it may be the result of something else: in this case a conscious mind developing and expounding a set of ideas in language. We can never be absolutely sure that it was not produced by a monkey randomly hitting typewriter keys, but we can calculate that the chances of its random production are so fantastically small that they approach zero.

If we take our monkey to represent chance or random variation, then we know that the likelihood of our lines of Shakespeare or hundred coin heads in a row being produced by chance are, to say the least, incredibly small. The risk of being in error because what we decide is a structure (Shakespeare, a row of heads) has actually been produced by chance is equal to this. We could refer to this as a type one error: a false rejection of a true null hypothesis that the event had occurred by random variation.