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From: brianu@ada-uts.UUCP
Newsgroups: net.puzzle
Subject: Re: random number wanted between neg inf
Message-ID: <11900003@ada-uts.UUCP>
Date: Thu, 19-Dec-85 12:59:00 EST
Article-I.D.: ada-uts.11900003
Posted: Thu Dec 19 12:59:00 1985
Date-Received: Sun, 22-Dec-85 01:12:11 EST
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Nf-From: ada-uts!brianu    Dec 19 12:59:00 1985


>***** ada-uts:net.puzzle / glacier!bhayes / 10:33 pm  Dec 16, 1985
>It may not make sense to talk about "choosing randomly" one of
>an infinite number of possibilities.  Just watch...
>
>Let's play a game.  I have, in this large and ornate paper bag,
>an infinite number of poker chips.  Each poker chip has one integer
>on each side, and, in fact, the two integers on any poker chip
>differ by exactly one.  The game is played as follows:
>
>1) I reach into the bag, and choose a random poker chip, without
>   peeking at the chip.
>2) I hold it between us so that I can see one side, and you can see
>   the other, but neither of us sees both sides.
>3) Either of us can reject the chip, just from seeing the side we see.
>4) We bet some fixed amount.
>5) The person with the low side wins.
>
>The distribution of the chips is as follows:
>Chip numbers     how many
>1/2              1
>2/3              2
>3/4              4
>4/5              8
>and so on
>
>I calculate the odds as follows:
>If I see "1" on my chip [a very unlikely event] I must win.
>For any other number [say "4"] there are twice as many chips which
>will win for me [8 "4/5" chips] as will lose [4 "3/4" chips].  Therefore,
>I should always bet, and I will win 2/3 of the time.
>
>You of course can make the same calculations and will also win 2/3
>of the time.
>
> -Barry "at no time do my fingers leave my hands" Hayes
>  bhayes@glacier

I don't see how you have proved your hypothesis, that it doesn't make
sense to discuss random numbers in an infinite domain.  First, the number
of points in the range 0-1 is the same as the number of points in the
range -oo to +oo. Secondly, lets play another little game:
  Since I don't have enough extra cash on hand to buy an infinite
number of poker chips, let's say we stop after the chips with 9 on one
side and 10 on the other.  Now let's play the game.  Like you said, if
I see a 1 then I know I've won.  If on the other hand, I see a 10 I know
I've lost. Now, for all the other possibilities the same reasoning as
above applies.  Suppose I see a 5, the other side of the chip could have
a 4 or a 6.  Since there are twice as many cases where I win (16) as where
I lose (8) you conclude that I have a 2/3 chance of winning. And of course
my opponent reasons the same way and decides HE has a 2/3 chance of
winning.  Since we both have a 2/3 chance of winning, by your reasoning
we must conclude that it makes no sense to discuss random numbers on a
finite domain.   Right.
    What this really means is that the reasoning is faulty.  The game can
be broken into two random acts: first the selection of the disk, second
the orientation of the selected disk.  Since each disk has a winning side
and a losing side, they are all equivalent. Thus only the orientation
matters.  Seeing the number imparts no information about this, so the odds
are just fifty/fifty, which is what you would expect.

Brian Utterback
Intermetrics Inc.
733 Concord Ave. Cambridge MA. 02138. (617) 661-1840
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