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What's the probability that two random US citizens have the same birthday. Is the probability 1/365? No, because birthdays are not uniformly distributed. Is it less or more? Here’s an unconventional solution to the problem subtly different from Ben’s (quoted; read his first) 1/ @ben_golub/1567018318917435393On twitter.com
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First we do ONE quick example: what if birthdays were EXTREMELY nonuniform and everyone’s birthday were January 1? Then the probability would be 1. So we heroically guess that if birthdays are nonuniform, the probability is > 1/365. Well, what if we’re wrong? 2/Permalink On twitter.com
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If so, then answering the original question would require empirical—rather than theoretical—analysis. And @ben_golub is a wicked smart theorist who likes to ask the kind of questions he could in theory answer while lying sleeplessly in bed with his eyes closed. 3/Permalink On twitter.com
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Mood +3 🙂
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So yeah, it’s almost certainly p > 1/365. (The argument that it’s a strict inequality probably requires some sort of appeal to epsilon-balls or trembling hand but Ben’s good at that stuff so let’s just go with it.) 4/Permalink On twitter.com
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Mood +2 🙂
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This is an awful technique early in one’s studies, because one doesn’t know enough about the skills/preferences of whoever’s asking the questions. But at some point, especially for questions where the person asking could look smart or make money from a particular answer… 5/Permalink On twitter.com
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…it can work really well. Lots of kids pass AP exams this way. 🤷 Note that “if X actually knew [or didn’t] how to answer question Y, that would be good for Z” (where X may or may not equal Z) is actually a core proof technique in economics and finance. 6/6Permalink On twitter.com
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Mood +3 🙂