Title: Structure, coincidence and chance
Course: UGQuants (SSPS) > Section: Probability

Structure, coincidence and chance

Social life is more structured than we sometimes imagine. There is an old adage that 'if you give enough monkeys enough typewriters, sooner or later one will produce the works of Shakespeare'. The underlying logic to that claim is that intelligence is, at least in part, a matter of luck and that structures pop up of their own accord from time to time. However, nothing could be less true. We can calculate the probability of a monkey writing Shakespeare at random rather easily and the results should serve as a reminder of how complex social structures (such as the grammatical structure of a language) are, and how far they are from being a matter of 'chance'.

If we include the combinations provided by the shift key and keys with punctuation marks, numbers, the space bar and so on (all keys, of course, are the same to our typing monkey!) there are approximately 100 possible alternatives for each key stroke.

Take the title of one of Shakespeare's many plays - The Tragedy of King Lear – which is only 24 characters long. Yet this short title represents just one of around 100,000,000,000,000,000,000,000,000 (10 to the power of 25) possible combinations of 24 characters. Even assuming we had enough typewriters and a population of monkeys large enough and diligent enough to produce ten million drafts every second, and that they had been working ever since the universe began, they would need to keep working for another 80 billion years or so to have an evens (50/50) chance of one of them randomly bashing out just the title, let alone the entire play!

The lesson is clear: even rather simple structures have very small probabilities of occurring 'by chance' or at random. However it is important not to confuse simple structure and mere coincidence. By the latter we mean the parallel occurrence of two similar events. We can use simple arithmetic to show that the probabilities of simple coincidences (as opposed to a pattern or structure) is often much higher than we might expect.

A group of people come together in the pub and are surprised to discover that two of them share the same birthday. 'Small world!' they proclaim. In fact the probability of two people sharing a birthday becomes greater than 0.5 (that is to say it becomes more likely than not) in groups of onlt two dozen or more people. (If you are interested, the calculation is fairly simple and is shown here under ‘Coincidences’).