The law of large numbers

Probability

Humans are good at attributing a cause but bad at guessing the likelihood of an event. A psychologist named Daniel Kahneman came up with an example. It is about a study of the incidence of kidney cancer in the 3,141 counties of the United States. The research revealed a remarkable pattern. The incidence of kidney cancer was the lowest in mostly rural, sparsely populated counties in traditionally Republican states in the Midwest, the South, and the West.1 So what do you make of that?

You probably came up with a few reasons why kidney cancer is less likely to occur in these counties, such as a healthy rural lifestyle or low pollution levels. But you probably didn’t think of randomness. Consider then the counties in which the incidence of kidney cancer is the highest. These counties were also mostly rural, sparsely populated, and located in traditionally Republican states in the Midwest, the South, and the West.1

The apparent contradiction can be explained by the fact that those counties all had small populations. And with smaller populations greater deviations from the average can be expected. Our intuition easily makes connections of causality but our reason doesn’t come into action to check whether or not it could just be randomness. We are inclined to think that some cause makes unusual things happen while these could just be random events.

In the summer of 1913 the ball fell on a black number twenty-six times in a row at the roulette wheel at the Casino de Monte-Carlo. Some people lost a fortune by betting that the ball would fall on a red number the next time. They didn’t realise that the chance of the ball falling on a red number never changed. The ball doesn’t remember where it fell the previous times. If we represent black with a B and red with an R, and assume for simplicity’s sake that there is no zero, it is possible to represent falling twenty-six times in a black number like this:

B B B B B B B B B B B B B B B B B B B B B B B B B B

The probability of the next twenty-six numbers being black is one in 67,108,864. That’s a long shot. What might surprise you is that the following combination of black and red numbers is exactly as likely to occur:

R B B R B R R B R B B R R B R R B R B B R R B B R B

You wouldn’t be thrilled if that happened unless you became a millionaire by betting on this particular sequence of twenty-six. And even then you didn’t think of the 67,108,863 sequences that didn’t materialise. We tend to consider only the things that did happen, but we rarely think of all the things that could have happened but didn’t. That might explain why events like the ball falling on a black number twenty-six times in a row impress us. And I am even more impressed because twenty-six happens to be my lucky number.

Try to imagine all what could have happened but didn’t happen. Imagine the probability of you sitting here and now reading this page, but as a prediction from 3,600 years ago. Imagine Joseph telling the Pharaoh: “I see (your name comes here) reading a pile of papyrus pages, not real papyrus pages, but images of papyrus pages appearing on something that looks like a clay tablet. It is named The Plan For The Future. But don’t be afraid, dear Pharaoh, for it will happen 3,600 years from now. But if we don’t set up this grain storage, it won’t happen at all, so we must do it. And by the way, Egypt will starve if we don’t.”

The odds for this prediction to come true weren’t one in 67,108,864, and also not one in 1,000,000,000,000,000,000,000 either. Even if you add considerably more zeroes to that number, the odds still remain far smaller. The probability is so close to zero that nobody can tell. Nevertheless you are sitting here reading this text. So how could this happen? The answer to this mystery is that so many things could have happened but didn’t happen, but something had to happen, and that’s what happened. It would have been impossible for Joseph to make this prediction unless the future is predetermined.

The licence plate on Franz Ferdinand’s car

So what to make of the reference to the end date of World War I on the licence plate number on Franz Ferdinand’s car? Franz Ferdinand was killed in his car and the assassination triggered the war. Some chance event helped the perpetrator. Franz Ferdinand’s chauffeur took the wrong turn after three conspirators had already failed. This gave him the opportunity to strike. He was hindered by the crowd surrounding him so he couldn’t aim very well. Nevertheless he managed to kill both the archduke and his wife with just two shots. This sequence of events was already remarkable.

The licence plate number makes it even more inconceivable. It might be possible to guess the end date of World War I by chance if you know when it starts. If you assume that the war wouldn’t take longer than twenty years, a random guess of the end date would be right one in 7,305 times. But something doesn’t add up here. The assassination succeeded after a series of mishaps, so if it were a prediction that accidentally turned out right, it would also imply a prediction of the assassination succeeding, Franz Ferdinand being killed in this car, and it being the trigger for the first world war.

That’s hard to do. And so Mike Dash in the Smithsonian noted: “This coincidence is so incredible that I initially suspected that it might be a hoax.”2 And because it isn’t a hoax, investigative minds should have probed other options. Conspiracy theorists didn’t take notice either, even though this incident fits into their schemes.

There is a story about a Freemason named Alfred Pike, who allegedly disclosed a secretive plan to bring about the New World Order and predicted both world wars with uncanny precision already in 1871. Alas, nobody ever heard of this plan before 1959. It is hoax. In the Netherlands they call it a monkey sandwich story. The licence plate number could have added some credibility to it. But then again, the truth is overrated. It matters more what people believe.

Seeing meaning when there isn’t any

“Everything is just random,” some pundits are eager to explain, “but because your mind is wired to see meaning, you see meaning. AIII 118 is just a random sequence of characters, but you attached meaning to it.” This book might be a random sequence of characters too, and yet you think it isn’t. Others might argue: “The language of Austria is German. Armistice in German is Waffenstillstand, so why doesn’t it read WIII 118, or even better, W 11 11 1918?”

If someone gives you a message, you don’t quibble about such details. If I say “hello” to you, you are not going to discuss with me why I didn’t say “hi” instead, unless you are a philosopher with a lot of time on your hands. Great Britain, the United States and France, which were all major participants in the war. These countries use the word armistice. It might be better to ask yourself how many sequences of characters with a length of six to eight are possible, and how many of them could refer to date of the armistice ending the war? That’s only a small portion for sure.

The law of small numbers

Everything is random and weird coincidences happen by chance. This is the law of large numbers. Pundits use the birthday problem to demonstrate that weird coincidences happen more often than we think. If you happen to share a birthday with another person in a small group, it might strike you as odd, but the chance of someone sharing a birthday with another person is already 50% in a group of 23. What they don’t tell you, is that the chance of you being one of those persons is a lot smaller. Weird coincidences are likely to happen, but less likely to happen to you. So if they happen to you all the time, it would be hard to explain as mere randomness.

And the law of large numbers may not apply to the licence plate number on Franz Ferdinand’s car. It applies to large numbers. How many historic events are out there that equal the importance of the assassination of Archduke Franz Ferdinand, the Armistice of 11 November 1918 or D-Day? The answer probably is: “Not many.” It is less likely that meaningful coincidences happen to such major historic events. To make it even harder to believe, the licence plate number coincidence not only implies a prediction of the end date of the war, but also the success of the assassination attempt, and this event being the trigger for the war.

Only a few historic events equal the importance of the assassination of Franz Ferdinand and the end of World War I. Perhaps this is just randomness like the incidence of kidney cancer varying wildly in small population samples. There are only a few historic events of similar importance. But D-Day is one of those few events, and the scheme surrounding D-Day is even more puzzling. This is a like four people out of a population of six suffering from kidney cancer and this population being the royal family of the country. Perhaps this is just randomness, but an experienced physician would consider that it runs in the family.

The fall of the Berlin Wall in 1989 was predicted. The coincidences surrounding the terrorist attacks of 11 September 2001 are truly dumbfounding. So if you are God, and you want your minions to notice, then what are your options? Framing the question like this makes the answer appear obvious. Indeed, there are countless other options, but asking why this particular path is chosen is as meaningless as asking why I said “hello” instead of “hi”. If you took a certain course of action to a certain aim, there are countless others you didn’t take. So if God wants us to take notice, we live in interesting times.

1. Thinking, Fast and Slow. Daniel Kahneman (2011). Penguin Books.
2. Curses! Archduke Franz Ferdinand and His Astounding Death Car. Mike Dash (2013). Smithsonian. [link]