Hand is well qualified for the task, being an emeritus professor of mathematics and former President of the Royal Statistical Society. But his book is written for readers who lack any special mathematical expertise and contains only a few simple artithmetical illustrations. He illustrates his Improbility Principle by means of a number of 'laws., so we get the Law of Inevitability, the Law of Truly Large Numbers, the Law of Selection, the Law of the Probability Lever, and the Law of Near Enough. This is perhaps a bit artificial but it serves as a basis for discussion.
The 'law of inevitability' says that if you make a list of all possible outcomes, one of them must happen. As Hand remarks, this is completely obvious, but we often forget it for that very reason. It means that if alll the possibilities are highly unlikely, one of them must nevertheless occur.
The 'law of truly large numbers' says that if the range of possibilities is large enough an unlikely outcome will occur sooner or later. He provides a striking example of this: if you choose any abitrary sequence of six numbers, such as the date of your birth in days, months, and years, you will find it somewhere in the expansion of pi; he gives a link to a site where you can try it out.
The 'law of selection' says that you can make probability as high you like (certain,in fact) if you choose after the event. Here he gives the example of scoring a bullseye by shooting an arrow at a barn door and then drawing a target round the arrow.
The 'law of the probability lever' says that a slight change in circumstances can have a huge impact on probabilities. The earth appears flat to us, but if we keep going in one direction we will eventually end up at our starting point.
The 'law of near enough' says that we tend to regard events that are fairly similar as identical—we over-interpret near coincidences.
An important reason for Hand's writing this book was to show that apparent coincidences are not a reason to believe in miracles, prophecies, or the paranormal. I doubt that any believers in these things will be persuaded by his arguments to change their views, but sceptics who are already disbelievers will find rational justification here for their scepticism.
17 May 2014