A simple betting problem was RE: my objection to the Doomsday argument

From: Rafal Smigrodzki (rafal@smigrodzki.org)
Date: Wed Apr 30 2003 - 12:50:14 MDT

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    Eliezer wrote:
    > Rafal Smigrodzki wrote:
    >>
    >> You know that half of all civilizations die at age 10 billion
    >> persons, and the other half live to 10 trillion persons.
    >>
    >> You know that you are a sentient belonging to one of these
    >> civilizations. There is a sealed envelope in front of you,
    >> containing your birth rank.
    >>
    >> What is the a priori probability of being in the doomed civ?
    >>
    >> 10e9
    >> P(X)=------------- ~ 0.001
    >> 10e11 + 10e9
    >>
    >> Your a priori probability of being doomed is small, because you know
    >> that most people live in the happy majority
    >>
    >>
    >> You open the envelope, find the number 999 999 999, and update your
    >> probabilities:
    >>
    >> (1-.5)1/10e9
    >> ----------------------------- ~ 0.99
    >> (1-.5)*1/10e9 + (1-.5)*1/10e11
    >>
    >> Your a posteriori probability of being doomed if your birth rank is
    >> 999 999 999, is 99%. Please note, that you didn't move a lightspeed
    >> from one civilization to another, merely adjusted your point of view
    >> based on the data that became available.
    >
    > Er... Rafal, shouldn't that be:
    >
    > > (1/10e9)*10e9
    > > ----------------------------- ~ 0.5
    > > (1/10e9)*10e9 + (1/10e11)*10e11
    >
    > I don't know where you're getting the (1-.5) from, since it's neither
    > a prior nor a conditional probability. And the end answer is, of
    > course,
    > 0.5, which corresponds to the reality that half of all people with
    > birthrank 999,999,999 live in short-lived civilizations, and half
    > live in long-lived civilizations.

    ### At first I did come up with .5 as the answer, but it seemed too simple,
    and I chose to get confused :-) Thanks for pointing this out.

    Talking about simple probabilities, here is a problem which initially
    baffled me:

    You play a game on TV. There is a large prize behind one of three doors. You
    are given a chance to bet on opening one of them. After you choose, the
    game's moderator will open one of the remaining two doors, an empty one. You
    can now change your bet, choosing the door that has not been opened, or you
    can stick with your initial bet. The question is, what should you do:

    a) stick to your previous choice
    b) flip a fair coin and either stick to your choice or choose the one
    remaining door
    c) always choose the one other remaining door

    It's really simple.

    Rafal

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