Re: OT: Re: Are you a "Mental Gymnast"?
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.
Let's have one more crack at this, so maybe you can see where
you are falling down here.
The rules of the game are:
PART 1: You choose one of three doors.
PART 2: Monty opens a door that (a) you did not choose and (b) has
no prize - Monty MUST open a door and it MUST have no prize,
and he MUST NOT open the door you chose. Therefore his choice
of door is NOT random.
PART 3: You stick with your choice or switch. You have additional
information, so your choice of door is NOT random.
You will grant (I hope) that if we ignore parts 2 and 3 then you
have a 1/3 chance of being right. If there are no other parts of
the game then one time in 3 you would choose the correct door.
Here is the tricky part, read this slowly and memorise - nothing
that happens before or after Part 1 alters the fact that when you
make your choice in Part 1 you had a 1/3 chance of being right.
Now - what if Monty said you can choose 2 doors? Then you would
have a 2/3 chance of being right, no? Well, essentially, by allowing
you to switch, Moty is letting you choose 2 doors.
Keep in mind - the only "random" action here is in part 1, when
you choose one door. Parts 2 and three are not random, because
the decisions made there are made with information to hand. Monty
is not allowed to choose a door at random.
Lets look at the steps again.
After PART 1 - you have a 1/3 chance of choosing right the first time.
After PART 2 - you still have a 1/3 chance that you chose right the
first time. Monty's action does not alter the fact that there is
a 2/3 chance that the prize is behind one of the doors you did not
choose. It is just that now you know which of the doors you did
not choose the prize is behind in the 2/3 chance that you picked wrong
the first time.
After PART 3 - you still have a 1/3 chance that you chose right the
first time. If you stick, there is a 1/3 chance you will win.
The only other possibility is that the prize is behind the other
closed door, and since the probabilities must add up to 1, there is
a 2/3 chance the prize is there.
I can tell you are still not convinced. What if the game plays in
a different order, like this:
PART 1: You choose a door (1/3 chance you are right).
PART 2: Monty lets you choose both of the 2 other doors. (2/3 chance
you are right - you have chosen 2 doors.)
At this stage Monty could open both doors you picked, and you would
have a 2/3 chance of being right, but to increase suspense it goes
like this:
PART 3: Monty shows you that the prize is NOT behind one of the doors
you picked. (there is still a 2/3 chance that one of the 2 doors you
picked has the prize.) Remember, Monty did not choose which door
to open at random because HE KNOWS WHERE THE PRIZE IS!
Finally, Monty opens the 2nd of the doors you picked - and 2 times
out of 3 that's where the prize will be.
The thing is - both of the above games are the same! If, in the
first game, you pick a door in part 1 and flip in part 3, it is
the same as if you chose 2 doors in part 1.
DavidP
--
David Purdue, Service Account Manager
Sun Microsystems Australia Pty Ltd
Ph: +61 3 9869 6412; Fax: +61 3 9869 6288; mailto:david -dot- purdue -at- sun -dot- com
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Follow-Ups:
- Re: OT: Re: Are you a "Mental Gymnast"?, Kevin McLauchlan
References:
OT: Re: Are you a "Mental Gymnast"?: From: Jan Henning
Re: OT: Re: Are you a "Mental Gymnast"?: From: Kevin McLauchlan
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