Consider the case where n=5. If n is 0, 1, 4 or 5 then this strategy will work. By simple binomial calculations, this will occur with probability 0.375. In the remaining 2 cases (each with prob 0.3125), the strategy only works when all three people in the majority guess there hat correctly at random. For n=5, this occurs with probability 1/8. Using Law of Total Probability to combine these we get P(Success) = 0.45.

The strategy only works more than half the time for n > 9, and for n=41 the success rate is 0.75. (I confirmed this with my own simulations).

Still, an interesting article.

]]>Seriously though nice article I couldn’t figure it out. Your solution reminds me of Kant’s categorical imperative.

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