ELi5: what’s the birthday paradox ? And is its mathematical basics a correct proof?
In: Mathematics
Human are *terrible* at probability and statistics, which is why the fields have many paradoxes. These are problems that we intuitively think one answer, but careful calculation show a different answer. Human are just bad at this kind of estimation. Here a paradox just mean a result contrary to intuition.
Whether they explicitly think in term of probability or not, most people would intuitively think that the probability for the birthday problem is exactly 1-((365-22)/365) ((365-21)/365) …. ((365-1)/365), which…is a correct answer.
The hard part is actually estimating this number, to see if this is at least 50%. This is the main difficulty, because ((365-22)/365) ((365-21)/365) …. ((365-1)/365) is a product of many factors, each of which are almost equal 1 (for example, the first factor is 343/365 which feels like “basically 1” to most people), so it doesn’t seem plausible that they multiply to a number <0.5. But here we actually have enough of these “almost 1” factor that we multiplied to a number <0.5. Which is why it is a paradox.
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