Yesterday was my grandfather’s 105th birthday party. He actually turns 105 tomorrow, on the 23rd of May. Yesterday was also supposed to be the end of the world, according to a very small splinter group of Christians. Clearly, the guy who made that prediction was thoroughly convinced, and thoroughly wrong.

Why do people make predictions like that? Let’s see what the math says! But first, more about my grandfather’s birthday party…

I was asked to give a speech at my grandfather’s party. At the end, I made the following observation.

- Take the day of the month my grandfather was born, 23.
- Multiply by the year he was born, 1906 (this gives 43838)
- Square this number (this gives 1921770244)
- Multiply this by the number of the month he was born, that is, 5 (this gives 9608851220)
- Multiply this by 38349 (this gives 368489835435780). This is not some arbitrary number, it is his exact age in days on the date of the party, 21st May. Here’s a nice exercise for you to try – check my working out!
- Finally, divide this whopping big number into the cube of his age (105 cubed is 1157625)
- The answer you get (1157625 divided by 368489835435780 is 0.000 000 003 141 538) equals Pi Over 1 Billion to 13 decimal places!!

So I presented this fact as some kind of conspiracy – a conspiracy to wish my grandfather the happiest and most memorable birthday possible.

I got different reactions from different people.

- Most people thought it was funny – but they weren’t sure if the formula was real.
- My grandfather commented “Only a Hartley could think of that”, which is not actually true.
- My uncle, also a mathematician, claimed I’d got my sums wrong, that it was only accurate to four decimal places – he’s partly right, in that the 0’s before the three don’t really count.
- Nobody, to my knowledge, attached any
*real*significance to the fact that some numbers related to his birthday combine to give*almost*pi.

So, *is* there any real significance? And what does this have to do with doomsday prophecies?

First of all, note that I didn’t get pi, I got pi divided by a billion. If you pick two random numbers out of the whole range of all possible real numbers, you can always multiply or divide one by some power of 10 to make it close to the other – to make 1906 close to pi, you divide by 1000, to get 1.906. That’s close to pi. Ok, it’s not *very* close, but it’s closer than 1906.

So suppose I take some other random number, and divide or multiply by some power of 10. How close can I *expect* to get to pi, or any other number? For example, what’s the chance that my scaled number equals pi to four decimal places? The answer is that it’s about 1 in 10000. Not exactly, but we don’t need exact probabilities here. A ballpark will do. To get five decimal places, I need a 1 in 100000 random number.

These probabilities are important. Statisticians have a way of testing ideas. Suppose someone proposes a radical idea (“There’s a birthday conspiracy!”) against the normal idea (“No conspiracy, sorry”) and presents some data to prove their idea (“Look! Pi Divided by 1 Billion!”). Then the statistician says “if the normal idea is true, how remarkable is this data? What’s the chance we’d see something like this anyway?” A 10% chance is not very remarkable, and shows that the data is no reason to believe the radical idea. A 0.001% chance is a different matter – maybe there’s something to this radical idea after all. The “gotcha” is that you have to calculate the probabilities properly.

If I had cooked up that one single formula – and that formula alone – and found it gave a number close to pi, that would be a remarkably lucky coincidence (or a sign of some cosmic conspiracy centering on my grandfather’s birthday party). However, if I tried two formulas, the chance is doubled. If I tried 10 formulae, the chance of such a coincidence is multiplied by ten.

So how many formulas did I try? I’ll tell you. I got a computer to try over 30000 different formulas. The chance of me hitting pi to four decimal places was about 95%. And if I somehow missed pi, there are other special numbers to choose from – pi squared, cubed, the square root of pi, pi plus 1, plus 10, and so on and on. Or instead of pi, I could look for other less-well-known special numbers, such as e, ln(2) or the golden ratio. All in all, it was a certainty that I would find *some* formula that worked, but it sure sounds remarkable when you hear it at the end of a speech and don’t know about the 29999 failed formulae.

This game shows that you can make a wide range of numbers given only a few “seed” numbers and a free choice of arithmetical operations.

Now, let’s talk about doomsday prophecies. The Christian radio host, Harold Camping, who so boldly proclaimed the end of the world yesterday apparently did some calculations from numbers in the Bible. I don’t know how many calculations he did or what kind, but he probably didn’t run the numbers through a computer program, and humans don’t have the patience to try thousands of formulae in one go. No doubt he saw some remarkable coincidences in his calculations, which made him bold enough to proclaim that the end of the world was the 21st of May 2011 – despite already being very publically wrong once before.

If he had applied the statisticians test, maybe he would have reasoned like this – “if there is no doomsday message here, I’m looking at a 1 in 10000 chance. However, this is the 20th time I’ve run the numbers, so it’s only a 1 in 500 chance. That’s still pretty amazing. Maybe the end of the world really* is* nigh?”

Unfortunately, there’s a fatal flaw in his probability calculation. I wrote about this flaw when I wrote about Paul, the FIFA-World-Cup Predicting Octopus. The Camping should have asked “How many other people are also doing similar numerical experiments?” The answer, in a world of 7 billion people, is probably “quite a lot!”

Then, he could have reasoned “If there is no doomsday message here, I’m looking at a 1 in 10000 chance. However, this is the 20th time I’ve run the numbers, so it’s only a 1 in 500 chance. However, there are probably 1000 other people running the same experiments, so this is not a 1 in 500 chance, in fact it’s almost certain that someone would come up with a numerical coincidence like this.” Then, he’d have broadcast somewhat more boring segments last week. He’d also be better prepared for today’s broadcasts, and somewhat less embarrassed.