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Math. proof of the superiority of independent guesses?

by Milton1 min read19th Jul 20207 comments

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Alice, Bob, Chris, Dana and Erica have set out to guess the number of candies in a glas jar just by looking at it. How do I mathematically show that the average guess will come closer to the real number if they guess independently, i.e without Bob hearing Alice's guess, Chris hearing Alice's and Bob's guess and so on?

And finally, I'm pretty confident I've read about this on LessWrong before, but I can't seem to find the specific post. Can anyone help me recollect?

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If the guesses are unbiased, the law of large numbers can be used to show this:

https://en.wikipedia.org/wiki/Law_of_large_numbers

(If you look at the proof, you can see where the independence assumption comes in.)