Suppose Jane wins the lottery three times. A person could say "well the chances of some person winning the lottery three times in the entire history of the world is expected by chance. No need to consider that it is rigged." But the prior number of lotteries is irrelevant to whether it was specifically likely for Jane to have won the lottery three times. There is a principle in philosophy called the principle of total evidence, where you have to consider all the evidence.
In this case, we didn't just observe that "some person won the lottery three times". We observed that Jane did. But is this principle accurate? And why should it be followed? Let's say O = Jane won the lottery three times and O' = someone won the lottery three times. Let H = chance hypothesis. When evaluating whether chance is at play, do we look at P(H|O) or P (H|O')? Clearly, the latter is higher, if not almost 1, considering how many lotteries have been played. Yet the probability of the first is miniscule. Yet still, something about generalizing the evidence here still feels right.
I see the same reasoning by people when considering coincidences, where they use the law of large numbers. "Given enough time and events, some improbable meaningful events will still happen" is the response to someone experiencing a meaningful coincidence. Yes, this is true. But this is not actually relevant to a specific observation that happens of course. If you see a coin land on heads 25 straight times, most would not immediately think "well, this had to inevitably happen by chance, since trillions of coins have been tossed, and sooner or later, 25 had to land on heads". But why is that wrong?
To me, sometimes, it does seem like the most relevant question is to ask if an alternative hypothesis better explains the specific (and not general) observation than the chance hypothesis. But I'm not sure about this. For example, I think it would be rational to think that seeing a coin land on heads 30 times means it's biased towards heads (say it's a p = 0.9 bias), even if the chance of atleast ONE coin in the history of coins landing on heads 30 times may as well be close to one (and possibly higher than 0.9)
