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But are there more farmers named Paige or teachers named Paige?
I can’t imagine Paige is a common name in many of the countries which still rely on subsistence farming, where farming will be a far more prevalent occupation. In the US, where Paige is a relatively common name, there are around twice as many teachers as farmers according to my very brief (and probably not super accurate) research.
Also I imagine that worldwide, farmers will skew male more than female. Just like how teachers probably skew more female than male. Note I didn’t bother to look for statistics for this, this is just a guess.
If you were to not name a person or gender and just say “is this person more likely to be a teacher or farmer,” then sure, farmer. But we’ve limited our base group of people to women named Paige. Surely that adjusts the probability.
Even if her gender and name adjusts the probability somewhat, it should all still be cancelled out by the order of magnitude by which the number of farmers outnumbers teachers, even in countries where a name like Paige is common. Teachers are an incredibly uncommon profession in the grand scheme of things, because it only takes a small proportion of them for a society to function (no offense intended to teachers of course)
But I will admit, I believe the original version of this thought experiment didn't give the woman a name. I was just trying to be a bit more descriptive haha. I'm pretty sure the thought experiment also predates the level of automation that is now common in agriculture. I was mostly just reconstructing it from memory.
Edit: yeah I just looked it up and in the US there is roughly the same number of teachers and farmers (around 3.5-4 million). I guess this particular version of the thought experiment is dead. But you could construct a similar one with a more common profession. Or just pretend it's still the 60s when you give your answer. Farming has simply become way too automated in the modern era.
Edit 2: I'm just now realizing the original version of this that I heard was with librarians, not teachers! Librarians are actually still orders of magnitude less common than farmers so it would work in that case. This is what I get for relying on memory for my clever comments.