1) The Monty Hall Problem (to be precise it hinges on ambiguity and the distinction between prior and posterior probabilities, but that is something most people aren't aware of and will get wrong the first time they see it, even people with knowledge of say Bayes' Theorem)
2) For several others, see Alon Amit's superb Quora answer to "What are the most interesting or popular probability puzzles in which the intuition is contrary to the solution?" ([2], login-walled). Mentions the very counterintuitive Penney's Game [0].
3) Berkson's Paradox, aka "People in hospital/getting treatment tend to have worse health indicators".
4) Asymmetric dice behavior is counterintuitive, when you first see it.
5) Benford's Law, on quantities occurring in nature (e.g. river lengths), as opposed to uniform distribution.
6) There are lots of counterintuitive things about Platonic solids.
7) Bayes' Theorem itself, superbly useful but possibly one of the things in probability most abused on a daily basis by bad journalism and bad statistics.
8) The Multiple Testing Problem/p-hacking/aka the xkcd "Green jelly beans cause acne"
and as a corollary:
8a) Most published (academic) findings aren't replicable, aka "Why Most Published Research Findings Are False", Joannidis (2005)
Great list! But I'm not sure the relevance? The intended point, which perhaps wasn't clear, was that logic / pure math as well as readily testable empirical theories are exceptions to the "everything is biased!" claim. There are in fact things that we can know with certainty, and that provides a foundation for a broader set of knowledge. The absolute relativist position is not just boring, but doesn't reflect reality either.
2) For several others, see Alon Amit's superb Quora answer to "What are the most interesting or popular probability puzzles in which the intuition is contrary to the solution?" ([2], login-walled). Mentions the very counterintuitive Penney's Game [0].
3) Berkson's Paradox, aka "People in hospital/getting treatment tend to have worse health indicators".
4) Asymmetric dice behavior is counterintuitive, when you first see it.
5) Benford's Law, on quantities occurring in nature (e.g. river lengths), as opposed to uniform distribution.
6) There are lots of counterintuitive things about Platonic solids.
7) Bayes' Theorem itself, superbly useful but possibly one of the things in probability most abused on a daily basis by bad journalism and bad statistics.
8) The Multiple Testing Problem/p-hacking/aka the xkcd "Green jelly beans cause acne" and as a corollary: 8a) Most published (academic) findings aren't replicable, aka "Why Most Published Research Findings Are False", Joannidis (2005)
[9] Almost-integers
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[1]: https://en.wikipedia.org/wiki/Monty_Hall_problem
[2]: https://www.quora.com/What-are-the-most-interesting-or-popul...
[3]: https://en.wikipedia.org/wiki/Berkson%27s_paradox
[5]: https://en.wikipedia.org/wiki/Benford%27s_law
[8]: https://en.wikipedia.org/wiki/Multiple_comparisons_problem
[9]: https://mathworld.wolfram.com/AlmostInteger.html
[0]: https://en.wikipedia.org/wiki/Penney%27s_game
More: https://en.wikipedia.org/wiki/Category:Probability_theory_pa...