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Bill Graham Bill Graham is offline
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Default another puzzler

spamtrap1888 wrote:
On May 15, 5:14 am, Carey Carlan wrote:
(Don Pearce) wrote
:



On Fri, 13 May 2011 16:05:33 GMT, Carey Carlan
wrote:


(Don Pearce) wrote in
:


On Fri, 13 May 2011 08:09:11 -0400, "Arny Krueger"
wrote:


"Bill Graham" wrote in message

Soundhaspriority wrote:
"Suppose you're on a game show, and you're given the
choice of three doors: Behind one door is a car; behind
the others, goats. You pick a door, say No. 1, and the
host, who knows what's behind the doors, opens another
door, say No. 3, which has a goat. He then says to you,
"Do you want to pick door No. 2?" Is it to your
advantage to switch your choice?" The above is a famous
problem. I've left out the
attribution to give you a few minutes (or forever, if
you want) to enjoy it. Bob Morein
(310) 237-6511


When you pick door #1 you only have a 1/3 chance of
winning. But after you see that there is a goat behind
door #3, your chance of winning is 1/2, so I would change
doors and pick door #2. But I don't really know
why....It's just gambler's instinct with me.


After you know there is a goat behind door #3 and are given a
chance to guess again, there is a 50% chance the car is behind
door #1 and a 50% chance the car if behind door #2. Change your
choice or not, you have a 50% chance of being right.


Lets make it ten doors. You pick one, and get a one in ten chance
of being right. That means that the chances are 90% that the car
is behind one of the 9 doors you did not pick. You know for
certain that at least eight of those doors conceal a goat, so
when eight goats are revealed, you have no new information. The
chances are 90% that the car is behind one of the nine - only now
there is only one remaining to open.


One vital fact here is that the person doing the revealing knows
the contents of the doors and chooses to reveal only goats. Had
he been guessing too, and just happened to reveal only goats,
then yes, you would be down to 50/50.


Alternate:


You walk in with 8 doors already open revealing 8 goats.
The car is behind one of the two remaining doors.
Convince me that your odds are not 50% to find the car.


Why? That isn't what happens. Read again and try to follow,
particularly the last part, which is the vital proviso.


Why? Because at the point of the final decision, that's the
situation.
How do the preceding 8 steps affect the final step?

Just as in flipping coins.
Getting 5 heads in a row is 1/32.
But getting the 5th head after already getting 4 is still 1/2.


The big difference: In the Monty Hall problem there is only one "coin
flip". Only one random choice is made -- the first choice of a door.
In the coin flip situation, there are five coin flips, five random
choices.

Now, in contrast, if the car and remaining goats were randomly
shuffled after each goat door was revealed, then the situation would
be different. But in the MHP problem the car does not move.


The really interesting thing is that, even if the car does not move,
conditional probability theory says the odds have changed, and you should
switch doors.