If you want a good example of how randomness-blind people are, attempt the following: propose a game for some randomly selected group.
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Make it competitive, but open form. E.g.: "The person who gives me the best reason why they should have this money, wins it."
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But there's a catch. Secretly, and at random, you pick the winner before you start.
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Let everyone observe everyone else's performance, and afterwards you announce the winner and ask everyone why they thought that person won.
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Most people will in no way realize that the outcome was randomized, and you will get some very plausible-sounding, but false, explanations.
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