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Please Stop Saying that it is a Million-to-one!

I could never be a politician or work in the media. Boris Johnson has been recently saying that the odds of the UK leaving the EU without a deal is a 1 million-to-one. The media also continually asks the question … “Is it still a million-to-one?”, and he answers, “Yes!”. When I hear this, I felt a bit reassured, as the impact of a no-deal on our future generations should not even be thought about. A million-to-one, to me, just seems good odds for it not to happen. Perhaps I’m too trusting in the usage of maths, or sometimes forget that the general public sometimes struggles to grasp even the simplists of concepts around numbers?

But, the bookies — who are good at probabilities and maths — put it at even money. So, someone, somewhere, has got their sums wrong:

If I toss a coin, the chances of me getting it right is evens (1/1). If I toss again, I still have a 50/50 chance of getting it right, but it is now 1 in 4 (25%) chance of getting the two tosses correct. Basically, it is 1 in 2^n and where n is the number of tosses.

I have to go all the way to 20 tosses of getting a 1 in a million chance of getting it right (1 in 1,048,576, in fact). Go and try this for yourself, and see if you can predict all the coin tosses, and you’ll still be trying next year.

To show the odds, a typical football match carries odds of around 5/2 (3.5 in decimal odds) for a draw. We can estimate the number of draws (n) that we need to predict for 1,000,000-to-1 with:

3.5^n = 1,000,000

and where 3.5^n is 3.5 to the power of n. And as I do logs in my crypto work, and work in a university that hosts John Napier’s toward, we can take the inverse log to find:

n = log_3.5 (1,000,000) = log10(1,000,000)/log10(3.5) = 11

We would thus need to predict 11 draws for football matches this Saturday, in order to gain odds of around 1 million to one.

How can politicians get away with saying things like this? If Brian Cox or Adi Shamir ever stood as a politician, they would get my vote. I come from a scientific world, where if you say something, you have to prove.

Professor of Cryptography. Serial innovator. Believer in fairness, justice & freedom. EU Citizen. Auld Reekie native. Old World Breaker. New World Creator.

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