Re: OT: Re: Are you a "Mental Gymnast"?

[Kevin McLauchlan <kmclauchlan -at- chrysalis-its -dot- com>]

On Wednesday 22 January 2003 02:49, Jan Henning 
wrote:
> > Yes. My thought is that their answer (FAQ
> > item 2.4)  is horse-pucky.
> >
> > The fallacy is that their answer assumes some
> > sort of causative connection between the two
> > choices.
>
> There is a causative connection: By making your
> choice of which door to open, you influence the
> choice of which door the game master will open.
> (Think about it: If you pick a losing door in
> the first round, you actually force his
> decision.)

Think of it the other way around. The first round 
is not complete until Monty has used his special 
knowlege to negate the effect of chance... of 
your "choice". 

Your choice has meaning only within the first 
round. No effect of your choice is allowed to 
pass to the second round... the real round. 
Monty's function is to ensure that you end up 
standing in front of a pair of doors, ONE of which 
has a prize and one of which does not. There 
is a probability of 1 that this will occur. Every 
time.

> The reason why it is correct to switch doors is
> not intuitive, at least for the overwhelming
> majority of people, but switching is still
> correct. (If you don't believe it, you can
> build a decision table where you track each
> possible combination of choices and outcomes.)

There is only one outcome. Chance is not operating 
in the first round. It is only for show. Monty's 
special knowlege, and his action upon that 
knowlege has only one purpose: to ensure that 
your first "choice" has no effect on anything but 
your mental state in the second round.

You always face two doors, where one hides a 
prize and the other hides disappointment. 

> I think the real lesson of this puzzle is that
> our intuition may be wrong, even when we are
> very much convinced of it being correct. Or, on
> a more practical manner: Our intuition is not
> suited to dependent probabilities.

Nope. Monty ensures that there is no dependency 
between the two choices. The first choice is bogus 
because you are not permitted to have any 
information about its result. Only Monty has 
information about what's behind the doors, so only 
he is necessary in the first step. Your 
"participation" is entirely for show, and has no 
effect on the outcome.  The *entire* mechanics of 
the event is that Monty puts you in front of two 
doors, one of which hides the valuables for the 
real, fifty-fifty selection. 

Perhaps your attachment to the fallacy is in 
the feeling that one wrong door is somehow 
different or special from another wrong door.
Again, the first "choice" is a red herring, meant 
to build up some participatory suspense in you and 
the audience.  Monty makes your first choice 
irrelevant to the outcome, because HE can exercise 
either of the two options that ALWAYS force the 
real, final choice to be:
You, standing in front of two blank doors, with 
no information that favors one door over the 
other.

The fallacy is in failing to recognize that 
Monty's special knowledge and his subsequent 
action are what eliminate your first "choice" from 
having any effect on the final outcome.

Perhaps your decision-tree process would have 
some validity if chance were operating unhindered, 
at both stages of the event, but it isn't.
Monty, and his knowledge (explicitly not shared 
with you), are there to ensure that it does not 
work the way you wish it did. Again, you always 
end up in front of a fifty-fifty choice.

To believe otherwise is to believe in magic.
Now, if MONTY had no knowledge when he 
picked a door to eliminate... well, then we could 
talk....    <g>

To quote Lois Patterson:
"Technical writing tie-ins: What appears to be
intuitively correct may not be. How a situation is
explained affects your interpretation of it. 
Surface logic is not always sufficient."

True. In this case, it's a matter of emphasis and 
subtle mis-direction. The key -- whether it's 
MONTY opening one (KNOWN BY HIM to be bad) 
door, or the goat-herd host, opening 999,998 
(KNOWN BY HIM to be goat-hiding) doors is that 
the opened doors are irrelevant. The real choice
that actually matters is between two identical 
doors.

The key in any such situation is that your 
probablity calculations are leaving out the 
fact -- which is revealed in the scenario, 
but intentionally downplayed -- that the 
complete first transaction does not ultimately 
allow chance to have any effect on its 
*outcome*.

It *appears* that chance is involved, because 
you make a random initial choice. It *would*
be a chance event if your choice were allowed 
to affect the outcome of the first round. 

But it doesn't.  You can choose doors all day, 
and your words, gestures, grunts, wishes, etc., 
do not affect what is behind the doors nor your 
access to them. That's the key. Monty's job is to 
ensure that your first choice is meaningless, and 
to do so without revealing any additional info. 
As long as Monty does his job, there can never 
be a probablity of anything but 1 that the two 
doors for your real choice are one-of-each.

Look again.  The story has two rounds (not 
three, as you might imagine...).
The first round ALWAYS finishes with two 
doors remaining closed, one hiding a prize, 
and one not. It makes not the slightest difference 
how many bad doors there are, if the host always 
eliminates all but a single bad one. It makes not 
the slightest difference to the outcome of the 
first event whether you chose the good door 
or a bad door, because the host CHANGES his 
response to bring about an UNCHANGING outcome.
The fact that it always ends that way is 
indisputable. If it always ends that way, 
then chance is not operating on the end-result of 
the first operation. Therefore, probability 
calculations that assume chance are being 
improperly applied. 

MECHANICS (if you read this far)

Kids' card trick...  You, me, and a deck of cards.
I force you to a certain card.

I shuffle through the deck and toss the aces and 
the tens, 'cuz I just don't like 'em. But let's 
see which card you like best. With a more patter,
we somehow get a card out of the deck, and I get a 
peek at it. Say, it's the four of spades. You 
don't see it, yet.

Me: First, which would you like to start with, red 
or black?

You: Black.

Me:  Fine. We'll eliminate red. Now, choose 
between the black face cards and number cards.

You:  Face cards.

Me: Fine. We'll eliminate them. Of the remaining 
cards, the black number cards, select high or 
low.

You: High.

Me:  Fine. We'll eliminate them. Of the remaining 
cards, two to five, do you prefer higher or lower?

You: Um... higher.

Me: Fine. So we'll eliminate the lower ones. Of 
the remaining two, do you prefer four or five?

You: Five.

Me: And what's left?  ... big grin as I turn up 
the four of spades.

At every step, I varied my patter depending upon 
what you chose. If you chose black, and the card 
was black, I say "we'll eliminate the reds".  But, 
if you had chosen red, then I'd say, "so, we'll 
eliminate them"...  In other words, by varying my 
patter, I force us to MY choice on the next step, 
regardless of what you choose each time.

Back at the doors, Monty forces the first outcome, 
regardless of what you actually chose.

There's nothing you can do in the first round 
that can cause the second round to be anything 
BUT one "good" door and one "bad" door.

Wishing won't make it otherwise. Appeals to 
authority won't make it otherwise.

If you think you are seeing an effect carry over 
from anything that the contestant did in the first 
round, it's because you haven't stated the 
problem clearly enough.

/kevin

whew!







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