The Laws of the Improbability Principle

The Law of Inevitability

Something must happen

The law of inevitability says that one of the complete set of all possible outcomes of a random event must occur. So, to see this law in action, we need need to be able to list all the possible outcomes, at least in principle: the set of all possible lottery tickets that might come up, the set of all birthdays in a year, and so on.

In the 1990s Stefan Klincewicz put together a syndicate to buy all 1.9 million tickets in the Irish lottery when its jackpot rolled over to £2.2 million. If he had managed to cover all the tickets, he would be guaranteed the jackpot – or, at least, a share of the jackpot because there is always the chance that someone else would also buy the winning ticket.

As it happened, despite impressive logistic arrangements, his syndicate managed to buy only 80% of the tickets, and although they did hold the winning ticket, so also did one other winner (history does not relate how many tickets the other winner had bought).

The Law of Truly Large Numbers

The Law of Selection

You can make things as likely as you want if you choose after the event

One of the classic stories about unlikely coincidences, into which some people read hidden messages, is that of the parallels between the lives of the two U.S. presidents Abraham Lincoln and John F. Kennedy.

It’s well-known that both were assassinated, but what is less well-known is that both were killed on a Friday, and both in the presence of their wives. Both were shot in the head from behind. Lincoln was killed in Ford’s Theatre, while Kennedy was killed in a car made by the Ford Motor Company. Furthermore, each had a son who died while they were president – Lincoln’s son Willie and Kennedy’s son Patrick. Lincoln had a personal secretary named John, and Kennedy had one named Lincoln. Lincoln became president in 1861 and Kennedy in 1961, while Lincoln’s assassin, John Wilkes Booth, was born in 1839 (his Wikipedia entry has it as 1838; if this is correct, it is an example of the law of near enough) and Kennedy’s assassin, Lee Harvey Oswald, was born in 1939. Both Lincoln and Kennedy were succeeded by presidents named Johnson who, wait for it, were born in 1808 and 1908 respectively. They each had four children.

I’ve put this as an example of the law of selection, but it could also serve to illustrate other laws of the Improbability Principle. As I said in the book, it is when the laws act in concert that the Improbability Principle becomes truly striking.

The law of selection is manifest by the selective nature of the matches listed above. These were selected from a large number (indeed, an infinite number – allowing the law of truly large numbers to apply) of potential pairs, most of which did not match: the names of their mothers, the birth dates of their mothers, the heights of their wives, the dates on which they got married, whether they were bearded or not, the precise nature of their religious beliefs, and so on endlessly.

The law of the probability lever is also hidden in this example. The fact that they were both shot in the head is cited as a coincidence. But gun assassinations normally involve being shot in the head or the chest (you don’t often hear of someone who was assassinated by being shot in the foot). So, given that they were both shot, the probability that both were headshots is quite high. Likewise the fact that they were both killed in the presence of their wives. Spouses often accompany presidents, so the probability that their wives were present is not as small as it might seem. (One is tempted to ask whether, had they been shot when their wives were not with them, that too would have been cited as coincidental.)

The law of near enough is manifest in many ways in this example. Certainly they were both assassinated in office, but not on the same date, and indeed, not even exactly 100 years apart. Lincoln’s personal secretary was named John, not Lincoln (or, to put it the other way, Kennedy’s was named Lincoln, not Abraham). As the law of near enough says: if you can’t get an exact match, how about something similar?

One is tempted to take any two other historical figures and see how many matches one can make. Do this for enough pairs, and rely on the law of truly large numbers (after all, how many such pairs are there?), and one is essentially certain to find “coincidences” such as the Lincoln/Kennedy story.

The Law of the Probability Lever

Slight changes can make highly improbable events almost certain

In 2005 one of my graduate students and I flew to the U.S. to attend a conference.  We’d booked our seats separately, because we were flying to different destinations after the meeting. But we found ourselves seated next to each other.

My first reaction was, what an extraordinary coincidence! After all, 747s normally seat somewhere between 400 and 500 passengers, so the probability of us being placed next to each other by chance looked like about 1 in 450 (taking the halfway point between 400 and 500 to illustrate). But then, I thought, we were travelling in the same class, so the number of available seats was less than 450.  Also, few seats occur in isolation. Normally they are in pairs (e.g. along the sides of the cabin) or in larger groups (e.g. down the centre of the cabin), so you have to sit next to someone. Furthermore, many, perhaps most people don’t fly alone, so the number of single seats is far fewer than appears. Put all these things together and they dramatically change the probabilities: instead of it being just 1 in 450 that we would happen to be allocated the seats next to each other, it becomes much more likely.

(I’m discounting the possibility that my student saw an opportunity at last to pin me down so that he could discuss his work in detail with me during the course of a flight lasting several hours!)

The Law of Near Enough

Events which are sufficiently similar are regarded as identical

In the 1950s, Eric W Smith, who lived in Sheffield, England was in the habit of collecting horse manure for his tomato plants from the woods behind the house. One day he saw another man doing the same. When he sat down on a bench to rest, the other man did the same. Eric introduced himself, saying his name was Smith. “So’s mine,” said the other man. So Eric expanded: “Eric Smith.” By then it was obvious that a strange coincidence was occurring: “And so is mine,” said the other man. “Eric W Smith,” said Eric. “Yes,” said the other man.

But further discussion revealed that the first Eric W Smith had the middle name Wales, while the second had the name Walter. So it wasn’t an exact match – but near enough to be surprising. But what if they had had different middle initials? Now not an exact match, but still surprising? How far can we relax the conditions for a match before we are surprised by it? How near does it have to be?

And again we should also look at the other laws of the Principle, to see if they are helping to produce this apparently highly improbable event. One thing we see is that Smith is the most common surname in the UK, with around 700,000 people sharing it – that’s about 1 in 100 people (this figure is nowadays: I don’t have the figure for the 1950s).