The latest (06 Oct 2017) Euromillion lottery numbers were 01 - 09 - 15 - 19 - 25 , with the "Lucky Stars " being 01 - 07.

Ha! Bet I convinced no-one about those numbers. The odds against 01 - 09 - 15 - 19 - 25 / 01 - 07 being the true lottery numbers are about 140 million to one, so I must have been a fool to think you'd believe me. The odds that I decided to lie is far more that one in a 140 million, so it's very unlikely those are the true numbers.

Wait a moment. Something is wrong here. That argument could be used against any set of numbers; and yet one of them was the real winning set. The issue is that probabilities are not being compared properly.

Let S=(01,09,15,19,25,01,07), let W(S) be the fact these numbers won the lottery, let L be the fact that I lied about the winning number, and L(S) be the fact that I lied and claimed S as the winning numbers.

So though P(L) is indeed much higher than P(W(S)), generally P(W(S)) will be higher than P(L(S)). That's because while W(S) gets penalised by all the other values S could have been, so does L(S).

Note the key word "generally". The sum of P(L(S')), across all S', is P(L); this means that for most S', P(L(S')) must be low, less than 140 million times smaller than P(L). But it's possible that for some values of S', it might be much higher. If I'd claimed that the winning numbers were 01 - 02 -03 - 04 - 05 / 06 - 07, then you might have cause to doubt me (after taking into account selection bias in me reporting it, the number of lotteries going on around the world, and so on). The connections with Komogorov complexity should be obvious at this point: the more complex the sequence, the less likely it is that a human being will select it disproportionately often (and you can't select all sequences disproportionately often).

What if I'd claimed that I'd won that lottery? That seems like a 01 - 02 -03 - 04 - 05 / 06 - 07 - style claim. I'm saying that the 01 - 09 - 15 - 19 - 25 / 01 - 07 did win (fair enough) but that these numbers are special to me because I selected them. Here you should be more sceptical. But what if I linked to a few articles that showed me with a gigantic novelty check? It would still seem more likely that I'd hacked and faked those than that I'd actually won... or would it?

Let's make it more

...

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