Re: reasoning under computational limitations

Eliezer S. Yudkowsky (sentience@pobox.com)
Sat, 27 Mar 1999 19:22:41 -0600

Wei Dai wrote:
>
> Suppose you wake up in a universe which contains a total of 20 people. Ten
> of them have been assigned numbers 0 to 9, and the other ten have been
> assigned the number equal to the 100!-th digit in the decimal expansion of
> PI. You are told your number but not anyone else's, and you are
> asked to guess the 100!-th digit of PI. Assuming that you can't actually
> compute that digit, it seems intuitive that your best guess would be your
> own number.
>
> My questions are (1) is this correct

Yes.

> (2) are there principles of
> reasoning under computational limitations (perhaps extensions of
> probability theory?) that can be used to derive or justify this and
> similar conclusions? Any relevant references would be appreciated.

Of course. There are eleven people with the correct digit and nine people with nine different incorrect digits. Ergo, your digit is probably the right one.

Furthermore, even if there were only four people and only one had the correct digit, it would still be correct to guess your own digit instead of picking a random different digit. That way your chance is one in four, while otherwise your chance is one in twelve - the probability you do not have the correct number already (3/4) times the probability of choosing the correct digit (1/9). 

-- 
        sentience@pobox.com          Eliezer S. Yudkowsky
         http://pobox.com/~sentience/AI_design.temp.html
          http://pobox.com/~sentience/singul_arity.html
Disclaimer:  Unless otherwise specified, I'm not telling you
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