Hey all,
I'd like to thank everyone who took a crack at the Let's Make a Deal
puzzle. I wish I could ceremoniously reveal the correct "solution" but I
don't know what is. Your guesses are as good mine (and in many cases,
better). Your various responses have kept me thinking and I'm still not
quite sure where I come down on it. Although everyone's input has been
helpful, I think the disagreement between Glenn and Lawry expresses the
problem well.
Let's kick it off with Glenn...
GLENN (as slightly edited by rick to reflect a 3 door game, just for
consistency's sake)
In the 3 door version of the puzzle, your initial guess has a
probability of 1/3 to have the good prize. All 3 doors have this
probability, and the sum of their probabilities add to 1 (as they must).
When Monty opens a door that shows a booby prize, the probability that that
door has the good prize drops to 0 (obviously).
RICK
Obviously. But this is where your account and Lawry's seem to part
ways. The question is now what happens to the probability that the
remaining doors have the prize.
LAWRY says
With the new information you now know that the good door is one of the
remaining two. While the odds HAVE changed, from 1-in-3 to 1-in-2,
switching your choice will neither increase nor decrease the odds of picking
the right one.
but
GLENN says (as slightly edited by rick to reflect a 3 door game, just for
consistency's sake)
...the sum of the probabilities doesn't equal 1 anymore (it would equal
2/3), and that is a no-no. Since the probability of your original pick is
fixed at 1/3, the other 2/3 of the probability has to be assigned to the
remaining door. Since there is only one other unopened door, the
probability of it containing the prize is 2/3.
RICK
The schism between Lawry and Glenn seems to revolve around the effect
that acquisition of new information has on the way the odds are calculated.
Or more precisely, whether the probability that your original pick is the
good door remains fixed, or whether it shares in the surplus created by
dropping the probability of the eliminated door to 0.... I'm not sure who to
believe here... I wish I had paid more attention in Stats 101.
Why should what happened in the first pick matter at all? Once Monty
moves on to the second pick, he is, in effect, asking you to choose again,
from scratch. But now with the benefit of the information that one of the
doors was wrong. You still have no reason to suspect that one door is
anymore likely to be the good door. What possible significance could the
fact that one of the doors is now being called "your original choice"
instead of "door #1" have? You still don't know where the prize is. How
are the odds any different than if Monty had just started with two doors and
asked you to choose between them?
Yet, Glenn's 100 door example seems pretty inescapable....
GLENN
Suppose there are 100 doors instead of 3. 99 have booby prizes and you pick
door 57 (because Heinz has 57 flavours :)). Monty then opens 98 of the
doors, each with a booby prize. (He can do this because he knows what's
behind every door.) Only door 66, and your original choice, door 57, remain
closed. He then offers you the opportunity to switch. By your reasoning,
door number 57 has switched from having a 1% chance to a 50% chance for
being the good prize because all those other doors are no longer an option.
That's not sensible. Obviously you would switch. There's a 99% chance the
good prize is behind door 66.
RICK
I would be interested in hearing Lawry respond to this.
And finally....
ERIN
Wierd you emerged from lurkdome. I just searched and emailed friends that
smurf quote you once posted because it was related to what we were
discussing.
RICK
If I've said it once, I've said it a million times... Never
underestimate
the intellectual appeal of blue gnomes wearing phallic hats.
take care,
rick
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