Toughest Game Show Question

D

divya bisht

Toughest Game Show Question
(Source : http://hardest-puzzle.blogspot.com/2012/01/toughest-game-show-question.html
)

You are on a game show and there are three doors. The presenter tells
you that behind one of doors there is a car and behind the other two
are goats, if you pick the car you win it. After you have picked a
door the presenter opens a different door with a goat behind it, he
then gives you the chance to change what door you open, what should
you do?

Discuss Solution in link below
http://hardest-puzzle.blogspot.com/2012/01/toughest-game-show-question.html

Follow Us on Facebook at
http://www.facebook.com/pages/Weekly-Hardest-Puzzle/277415275612476
 
B

Bill Reid

You are on a game show and there are three doors. The presenter tells
you that behind one of doors there is a car and behind the other two
are goats, if you pick the car you win it. After you have picked a
door the presenter opens a different door with a goat behind it, he
then gives you the chance to change what door you open, what should
you do?
OH NO, THE DREADED "MONTY HALL" PROBLEM, IT'S BEEN AT LEAST A
DECADE SINCE I'VE SEEN THIS MONSTROUSITY REAR IT'S UGLY HEAD IN
USENET!!!!!!!

The answer is that it doesn't matter, your odds are still one
in three no matter what you do...but wait, does it matter if the
presenter knows what's behind the doors and shows you the goat
to make the game more interesting? Let's see what happens with
the "regs" here, might be fun...
 
T

tom st denis

OH NO, THE DREADED "MONTY HALL" PROBLEM, IT'S BEEN AT LEAST A
DECADE SINCE I'VE SEEN THIS MONSTROUSITY REAR IT'S UGLY HEAD IN
USENET!!!!!!!

The answer is that it doesn't matter, your odds are still one
in three no matter what you do...but wait, does it matter if the
presenter knows what's behind the doors and shows you the goat
to make the game more interesting?  Let's see what happens with
the "regs" here, might be fun...

Never switch your bet. Pigeons know it.

Tom
 
E

Eric Sosman

OH NO, THE DREADED "MONTY HALL" PROBLEM, IT'S BEEN AT LEAST A
DECADE SINCE I'VE SEEN THIS MONSTROUSITY REAR IT'S UGLY HEAD IN
USENET!!!!!!!

The answer is that it doesn't matter, your odds are still one
in three no matter what you do...but wait, does it matter if the
presenter knows what's behind the doors and shows you the goat
to make the game more interesting? Let's see what happens with
the "regs" here, might be fun...

Stick with the original choice. You can't eat a car.
 
J

John Gordon

In said:
The answer is that it doesn't matter, your odds are still one
in three no matter what you do

It does matter. If you switch, you have a two-in-three chance of winning.
..but wait, does it matter if the presenter knows what's behind the
doors and shows you the goat to make the game more interesting?

The presenter does know. He doesn't open a random door.
 
S

Stephen Sprunk

OH NO, THE DREADED "MONTY HALL" PROBLEM, IT'S BEEN AT LEAST A
DECADE SINCE I'VE SEEN THIS MONSTROUSITY REAR IT'S UGLY HEAD IN
USENET!!!!!!!

The answer is that it doesn't matter, your odds are still one
in three no matter what you do...but wait, does it matter if the
presenter knows what's behind the doors and shows you the goat
to make the game more interesting? Let's see what happens with
the "regs" here, might be fun...

Mythbusters recently tested this. You're always better off switching.

S
 
S

Seebs

The presenter does know. He doesn't open a random door.

Our spammy friend seems to have a religious objection to an unambiguous
problem statement, and has not specified this.

-s
 
S

Shao Miller

OH NO, THE DREADED "MONTY HALL" PROBLEM, IT'S BEEN AT LEAST A
DECADE SINCE I'VE SEEN THIS MONSTROUSITY REAR IT'S UGLY HEAD IN
USENET!!!!!!!

One can download the following book "Information Theory, Inference, and
Learning Algorithms" by David J. C. MacKay from his site:

http://www.inference.phy.cam.ac.uk/mackay/itila/

Or better yet, one can purchase it and have it shipped.

This problem is discussed on page 57, exercise 3.8, if I'm not mistaken.

However, the above-stated details are a little different.

- Shao Miller
 
K

Keith Thompson

Seebs said:
Our spammy friend seems to have a religious objection to an unambiguous
problem statement, and has not specified this.

The question does say that the presenter opens a door with a goat behind
it.
 
J

James Kuyper

The question does say that the presenter opens a door with a goat behind
it.

Yes, but it doesn't say whether he opened a random door that happened to
have a goat behind it, or whether he deliberately choose a door that he
knew had a goat behind it. The latter is, I believe, the intended
situation in the standard "Monty Hall" problem. A lot of very confusing
discussions about that problem have been based upon choosing a different
assumption. To take the most extreme case: what if the presenter only
bothers opening the second door if you happen to have already chosen the
correct door, and otherwise leaves it closed (obviously, you've not been
informed of this policy)?
 
S

Seebs

The question does say that the presenter opens a door with a goat behind
it.

But it does not say whether we know that the presenter ALWAYS does this,
or whether it's just that, on this single case, the presenter HAPPENED to
do this.

Imagine that you were running a game show for people who are familiar with
this problem; the host strategy would be to always show people a goat if they
picked the car, and not show them a goat if they already picked one. Then
people who know the problem in the formulation where the host *must* always
show you a goat will end up losing every time, because they only get the option
to switch when doing so will hurt them.

In the real show, the host chose whether or not to reveal another door in
a way that made it not obvious that you were better off switching, because
part of it was his read of the contestants.

-s
 
B

Bill Reid

Mythbusters recently tested this.  You're always better off switching.
Did they use a Monte Carlo simulation where they fired cannonballs
through the garage doors of dozens of innocent people to see whether
there was a goat or car in the garage?

You're always better off switching, EXCEPT if the presenter
knows you've chosen the car and ONLY offers you a different
door to potentially keep you from winning the car. In THAT case,
you WILL win the car 1/3 of the time overall, and will win
100% of the time when you decline the switch...so the behavior
of the presenter DOES make a difference...
 
E

Edward A. Falk

Ooooh, I love that problem. I've made money on that one.
Never switch your bet. Pigeons know it.

That's why they're called "pigeons".

The key is that Monty knows where the goats are, and thus
introduces information into the game when he opens a door.
Thus, you should always switch. Math on request.
 
K

Kaz Kylheku

OH NO, THE DREADED "MONTY HALL" PROBLEM, IT'S BEEN AT LEAST A
DECADE SINCE I'VE SEEN THIS MONSTROUSITY REAR IT'S UGLY HEAD IN
USENET!!!!!!!

The answer is that it doesn't matter, your odds are still one
in three no matter what you do...but wait, does it matter if the
presenter knows what's behind the doors and shows you the goat
to make the game more interesting? Let's see what happens with
the "regs" here, might be fun...

I think you an easily solve this by a simple division into cases,
and simple probabilities.

Look Ma, no conditional probability, no Bayes' theorem.


Suppose Monty always reveals the goat. (This builds suspense in the
audience and is good for the show's ratings, which are more important
than whether or not cars are given away.)


There are two cases:

Case 1: (p = 2/3) You picked the goat.

Monty knows that this case is occurring, and reveals to you the other goat! So
you must switch to get the car, and by switching the car is guaranteed.

Case 2: (p = 1/3) You picked the car.

In this case, you must not switch; you're already on the car.
Of course Monty reveals a goat in this case also.


Now you don't know which of these cases you are in. But you do know
that one of these cases requires switching and the other requires
staying and that the one that requires switching occurs with p = 2/3, and the
one that requires staying occurs with p = 1/3.

This essentially translates to "the probability that you must switch in order
to win the car is 2/3", and that translates to "the probability that
the car is behind the door you did not pick in round 1 is 2/3".

If you do not switch, you are betting on being in Case 2, which is p = 1/3: the
same odds as a one-round draw.

If you switch, you are betting that you are in Case 1, where you have p = 2/3
odds, an improvement.



Switching: no brainer.
 
S

Stephen Sprunk

Did they use a Monte Carlo simulation where they fired cannonballs
through the garage doors of dozens of innocent people to see whether
there was a goat or car in the garage?

They did a Monte Carlo simulation, but unfortunately nothing was
destroyed in the process.

I prefer the mathematical proof, but that doesn't make for good TV.
You're always better off switching, EXCEPT if the presenter knows
you've chosen the car and ONLY offers you a different door to
potentially keep you from winning the car.

Their test scenario was that the presenter _always_ opens a losing door,
which they said is how the actual game show worked.

They also first tested how often people switched after that door was
opened; interestingly, zero (of twenty) contestants did so.

S
 
S

Seebs

Their test scenario was that the presenter _always_ opens a losing door,
which they said is how the actual game show worked.

They are wrong, I'm pretty sure. One of the discussions of this included
an interview with the guy, and there was no such rule.

-s
 
K

Kaz Kylheku

They are wrong, I'm pretty sure. One of the discussions of this included
an interview with the guy, and there was no such rule.

But odds are 2/3 that he lied in that interview.
 
J

James Kuyper

They are wrong, I'm pretty sure. One of the discussions of this included
an interview with the guy, and there was no such rule.

-s
<http://en.wikipedia.org/wiki/Monty_Hall_problem> contains a section
titled "extended problem description" which fills out the unstated
assumptions behind the original statement of the problem. There's a fair
number of them.
There's also a section titled "other host behaviors" which summarizes
the results when other assumptions are made.
 

Ask a Question

Want to reply to this thread or ask your own question?

You'll need to choose a username for the site, which only take a couple of moments. After that, you can post your question and our members will help you out.

Ask a Question

Members online

Forum statistics

Threads
473,734
Messages
2,569,441
Members
44,832
Latest member
GlennSmall

Latest Threads

Top