Taleb’s Fair Coin
In his ‘Black Swan’, Taleb has the following example:
[Taleb]: Assume that a coin is fair … . I flip it 99 times and get heads each time. What are the odds of getting tails next time?
You might like to think about it. …
Taleb finds that ‘nerds’ (including mathematicians of his acquaintance) tend to say 0.5, whereas others such as ‘Fat Tony’ (of whom he approves) say ‘zero’ or ‘almost zero’. This example seems to me to be at the heart of his negative views about mathematicians. It also reminds me of similar conversations among pupils on first encountering probability theory. What is the ‘mathematical’ answer?
In the school-room the word ‘assume’ is taken to mean ‘I may know this isn’t quite right, but just go with it for now’. Or at least ‘don’t challenge it’. Thus the nerd faced with an academic question will say ‘0.5’, supressing any quibbles.
On first contact with scholastic probability theory, it is easy to think of it as a topic within applied mathematics. In which case you would be expected to state any assumptions required, such as that all the tosses under consideration have or will be conducted under identical terms. But one wouldn’t get through many probability theory problems if you did this explicitly. Rather you ‘bear them in mind’. (Or possibly nerds don’t?)
The statement ‘assume that a coin is fair’ said of a real coin (as Taleb seems to intend) is interesting. Anyone with even the slightest acquaintance with probability theory will realise that no-one can ever know that any given coin is really fair in the sense of probability theory. Even if a coin has been tossed millions of times and the results are consistent with the hypothesis of fairness, this says nothing about the probability of heads when tossed differently. So, in defence of nerds, it seems to me that if ‘assume that a coin is fair’ is to mean anything, it should mean more-or-less what it means in a probability theory examination. On the other hand, Fat Tony seems to take it as a sign that fairness is ‘only an assumption’, and so disregards it in the face of overwhelming evidence to the contrary. That is, he treats it as he would any assumption in his world.
Which is the appropriate answer? It depends. At school I regarded this type of question as a school-boy trick. Whatever you answered the questioner would claim a different meaning of ‘assume’ and ridicule you. I’m sure I was right, especially if I was asking the question. To avoid criticism I used to give conditional answers: if you mean X then the answer is Y, but … . In my professional life I have found that people don’t always take kindly to me questioning their questions, so I tend to say (translated in Taleb-ese) something like:
The nerdish answer is 0.5. The practitioners’ answer will often be zero or about zero. But could it be that Taleb is skilful with coin tosses and playing a game with us, so that the probability is almost 1? We need to understand the context more.
That is, I present the range of ‘reasonable’ results based on the common approaches, and then present a scenario which gives a result that is as far as reasonably possible those that others have claimed. This is at least as ‘mathematical’ as the nerdish approach and – I claim – as much informed by experience as Fat Tony. It generally leads to a discussion of ‘assumptions’.