Wednesday, 27 June 2012

Winning the impossible lottery

Photo: Nick See
I wanted to call this post 'The Lottery Paradox', but it turns out that title is already taken by a very similar idea. The version I'm going to discuss is a bit more extreme and has a different focus.

It is inspired by some thoughts I've been having related to the validity of the Anthropic Principle (the subject of my last post), inspired by the comments left by Callum J Hackett on that post.

Suppose the nation of Foobar has a national lottery, but the way it works is that every single day, one of the citizens is chosen at random to win a big prize.

Now suppose that the nation of Foobar has an arbitarily huge number of citizens (say a Googolplex).

I'm going to try to persuade you to believe two completely opposing conclusions about the possibility of believing that you have won this lottery. One of these conclusions is true, and one of them is false. See if you can tell which is which.

You turn on your TV and see that you are today's lottery winner. Should you believe that this is true?

You should never believe that you have won

If the number of citizens is arbitrarily large, then the probability that you will win is arbitrarily small. It is effectively zero.

This means that it is essentially impossible for you to win. If you learn that you have won, then there simply must be some other explanation. It doesn't matter even when the money arrives in your bank account. It simply cannot be the case that you have won the lottery. No amount of evidence should convince you that this has happened.

Any other explanation that you can imagine, no matter how fantastic, is almost certainly more probable than that you have won legitimately. If you find yourself in such a situation, the rational thing to do would be to assume that you are hallucinating, or that some anonymous benefactor has arranged some elaborate conspiracy theory, or even that you are a brain in a vat and everything that you believe to be true is a computer generated illusion.

Now, you might as well carry on and behave as though you had won, especially once the money lands in your bank account, but no matter what evidence is presented to you, you should never ever believe that you have really won the impossible lottery.

You should believe that you have won

Every day, one of your compatriots is chosen to win the lottery. These people often react with disbelief, accepting the money but being convinced that there must be some other explanation.

You know that these people are incorrect. There is no conspiracy. Those people are certainly not brains in a vat, because you know yourself to be real. They must have fallen victim to some sort of logical error to have made such a mistake. You know you would not be so foolish were you ever to win the impossible lottery.

And then, one day, you do win. Naturally you are shocked and amazed, and initially don't believe what you see on the television. Surely you've made a mistake? But no, you check and double check, and sure enough the money appears in your bank account.

Recalling your opinion of those winners who have gone before you, you realise that from an detached point of view, it is not surprising at all that you have won. Any surprise is purely a result of your own subjectivity. Really, there is no objective reason why you should not be the winner.

You take your money and realise that you have been incredibly lucky. Unfortunately, nobody believes you when you tell them!


  1. Interesting! I have to say, I do not know which to choose. :P I'll have to comment in full with our other disagreements at the weekend - my exam is due Friday, so my attention must really be fully devoted to it now! :)

    1. Best of luck with the exam! Stay away from this blog and my emails in the meantime!

  2. I'll be posting my thoughts on it shortly, I just want to get a bit of a feel for what people think first.

    I'm pretty convinced I have the correct answer, but I'm far from convinced that I can explain it compellingly. I might have to bring in a mathematician or two I know to back myself up with arguments from authority!