Littlewood's law states that a person can expect to experience an event with odds of one in a million (defined by the law as a "miracle") at the rate of about one per month. This is attributed to the theory of truly large numbers.
I know that Littlewood's point of view is defendable. However, the question I want to ask is: Is this "law" regarded with seriousness in the mathematical community (read: is it a proven fact or is it just a hypothesis?), and is there scientific evidence of the phenomenon?
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$\begingroup$It's not really a mathematical claim, so mathematics as such take no position of it.
Almost the entire content of the claim depends on what exactly one considers an "event" worth caring about in this context to be. If I throw a fair coin twenty times and note down the result, then with mathematical certainty the odds of me having got that exact sequence of heads and tails will be precisely $1:1,048,575$. According to the premises of the "law" then, apparently me deciding to throw a coin twenty times will guarantee that I'm going to experience a miracle!
Or even without throwing coins, if it's raining and I look at the window, the odds of having exactly that distribution of raindrops with respect to each other will be well below one to a million.
But of course me getting THHHTTHHHTTTHHTHHHTT by flipping coins isn't really a miracle. That's an entirely humdrum and non-distinguished series of throws. And that's the problem with interpreting the claim.
What we really need to ask if we want to find out if the law is true is: What counts as an "event" here? And that is not a mathematical question.
Littlewood's law is then really a claim about psychology or perception rather than about mathematics. It claims:
Of all the trillions of possible happenings with one-to-a-million-chances, there are so many that would strike us as surprising if they actually happened and someone noticed, that a person will come across the unfulfilled possibility of one of them happening every few seconds on average.
(There are about 2.5 million seconds in a month).
But that is really a claim about what we're capable of being surprised by, not one of probability theory.
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