[REBOL] Re: My Statistical Thoughts on Monty Hall Problem (non-REBOL)
From: joel::neely::fedex::com at: 20-Dec-2001 8:02
Hi, Laurence,
[lgidding--aaa--allianz--com--au] wrote:
...
> You appear to believe that the original 1/10,000 chance is
> still valid after 9998 the doors have been opened.
>
He does. It is. See below.
> Surely like the coin which has no knowledge of its previous
> toss, the odds have no knowledge of their previous length.
>
> For example:
>
> Imagine a Roulette wheel with 10000 numbers
> I pick no 1 and have a 1/10000 chance
>
> The next time the wheel has reduced to 36 numbers
> I still have no 1 but now I have a 1/36 chance
>
The last sentence above is the key to the flaw in your
analysis/argument. You talked about changing the number
of outcomes from the wheel and then discussed the odds for
the next time
.
There is no "next time" in the Monty problem we've been
discussing. Your argument would only be relevant under
the following scenario:
* One round of the Monty game is played where the prize is
placed behind a door, the contestant picks a door, and
win/loss is declared. I think we all agree that the
probability of a win is 1/3.
* A door is removed from the stage, leaving two doors.
* A "next time" round is played, where the prize is placed
behind a door, the contestant picks a door, and win/loss
is declared, WHERE THE LOCATION OF THE PRIZE AND THE
CONTESTANT'S CHOICE ARE UNCORRELATED TO THOSE OF THE
"FIRST TIME" ROUND. I think we all agree that the
probability of a win is 1/2.
However, in the game we've been discussing, analyzing,
simulating, and argu^H^H^H^Hthinking ;-) about, there is
ONLY ONE ROUND, so all of the decisions about which door
Monty can open, and which doors remain available to the
contestant's keep-or-switch decision ARE ALL CORRELATED
TO THE UNCHANGING FACTS OF THE TRUE LOCATION OF THE
PRIZE AND THE CONTESTANT'S FIRST CHOICE.
It is precisely those not-so-obvious-to-the-casual-observer
correlations that make this game a good demonstration of
why everyday common sense falls down and skins its knees
so often over mathematical and probabilistic problems.
(Not to mention design and verification of programs! ;-)
Let me offer a counter-example to the line of analysis
you proposed. Suppose I have four coins (penny, nickel,
dime, and quarter). I place them in my cupped hands and
shake them, then separate my hands without letting you
see their contents. I tell you (truthfully!) that one
of my hands contains a single coin and the other hand
contains three coins. You have to guess which hand has
only one coin.
If we can assume that my shaking hands are an unbiased
randomizer, and that I am truthful, then I hope we can
agree that your probability of winning is 1/2.
Now, suppose I wiggle the fingers of one hand, say the
left one, and a coin drops to the floor from that hand.
Obviously this action changes neither the original
distribution of coins between my hands, nor your stated
guess as to which hand holds only one coin.
Now, suppose I wiggle the fingers OF THE SAME HAND, and
another coin drops to the floor from that hand. Again,
this action changes neither the original distribution of
coins nor your stated guess.
However, I hope we can agree that you can now state with
absolute certainty which hand originally held one coin.
The "state" of the system (WRT to which hand originally
held one coin and which hand you originally guessed) has
not changed, but YOUR INFORMATION ABOUT THE STATE *HAS*
CHANGED and may be used to guide any future decisions
you make WITHIN THIS SAME ROUND.
Someone may find it interesting to analyze (or simulate)
the probability of success if I allow you to change your
mind after the first coin has dropped.
-jn-
--
; sub REBOL {}; sub head ($) {@_[0]}
REBOL []
# despam: func [e] [replace replace/all e ":" "." "#" "@"]
; sub despam {my ($e) = @_; $e =~ tr/:#/.@/; return "\n$e"}
print head reverse despam "moc:xedef#yleen:leoj" ;
