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unJon
unJon
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January 16th, 2019 at 11:31:15 AM permalink
Quote: Wizard

What if the host knew where the car was and planned to only open another door if you picked it, thus tricking Bayesians into switching.

Fair. Itís like the old answer to Pascalís wager.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
netzer
netzer
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January 17th, 2019 at 11:09:44 AM permalink
Quote: DJTeddyBear

Never assume.

Monty doesnít always offer the switch, and he himself has said that he would often offer that option based upon his opinion of the contestant - and that included both the second chance to win the prize as well as the second chance to get zonked.


Since you frequently respond to math postings with "I don't understand the math but..." I conclude that you are not a mathematician, so let me explain:

In analyzing real-life situations mathematicians frequently employ mathematical models which may not perfectly model the situation but which are easy to solve or manipulate. They would rather have an imperfect model that is easy to solve than a precise model that is impossible to solve.

The Monty Hall Problem is just a mathematical problem and does not accurately represent the way Monty ran his show. To make it solvable one stipulates the following conditions:

1. Monty knows which door the car is behind.

2. Monty offers the switch whether the contestant has chosen correctly or not.

Only with these conditions is the problem solvable.

I get the impression that the Wizard still feels that Marilyn's answer was wrong. Well, he is in good company. When Marilyn published her solution she received a flood of letters from people with Ph. D.s and on academic letterheads telling her that she was wrong and begging her to recant. All of these people since have come over to her side.

Marilyn is too modest to present herself as having the world's highest I. Q. Her publisher does that. I used to belong to a high-IQ society (Mensa) and discovered that having a high IQ doesn't prevent some people from saying or doing stupid things. My favorite Marilyn stupidities are:

Q. I bought some stock and it has gone down. What should I do?
A. You have a loss only if you sell.

and

Q. If lost at sea, how do I find land?
A. Follow the waves, for waves always crash on a shore.

Nevertheless, I think that Marilyn contributes to society.
Last edited by: netzer on Jan 17, 2019
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Wizard
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Wizard
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January 17th, 2019 at 11:30:51 AM permalink
Quote: netzer

I get the impression that the Wizard still feels that Marilyn's answer was wrong.



No, I feel she was right for how she interpreted the question. I fault her for not clarifying what the question is asking. I found images of the original question and answer.

Question:



Answer:



You can see the question stated that the host knows what is behind the doors. However, he didn't say why he opened up a goat door or if he did this all the time. Marilyn assumed the host will always reveal a goat after the contestant picks a door. Under this assumption, her answer is correct.

However, many people interpret it like there was a door malfunction, revealing a goat. If that was the case, then there is a 50/50 chance the original door has a goat.

I fault her not for giving a wrong answer, but not clarifying what the question is.
It's not whether you win or lose; it's whether or not you had a good bet.
Ayecarumba
Ayecarumba
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January 17th, 2019 at 12:06:24 PM permalink
It seems Marilyn describes it like this, with 1/3 probability of each initial choice:


Initial Unknown Choice Monty Reveals You Swtich To Result
Car Goat 1 Goat 2 Loss
Goat 1 Goat 2 Car Win
Goat 2 Goat 1 Car Win


However, isn't it actually like this?:


Initial Unknown Choice Monty Reveals You Swtich To Result
Car Goat 1 Goat 2 Loss
Car Goat 2 Goat 1 Loss
Goat 1 Goat 2 Car Win
Goat 2 Goat 1 Car Win
Last edited by: Ayecarumba on Jan 17, 2019
Simplicity is the ultimate sophistication - Leonardo da Vinci
DJTeddyBear
DJTeddyBear 
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January 17th, 2019 at 1:45:39 PM permalink
Quote: Wizard

I've watched old shows on YouTube and he didn't offer a switch.

He never offered to switch DOORS. Doors were only used on the big deal finale and never had a zonk - just 3 prizes of widely varying value. Also, the big deal involved two contestants making a switch unfair.

Unfortunately, the problem is always worded using doors rather than curtains. Monty often - but not always - allowed a switch of the curtains (or boxes, etc.).


Quote: netzer

Since you frequently respond to math postings with "I don't understand the math but..." I conclude that you are not a mathematician...

Youíre right. Iím not a mathemetician per se, but this is simple math and I understand and agree with the Marilyn / math answer.

My point is, when you examine the tv show, you have to realize that Monty (and subsequent hosts) had their own reasons and parameters for occasionally offering the switch which adds to the unpredictability and could make the math invalid.
Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
netzer
netzer
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January 17th, 2019 at 1:49:04 PM permalink
Ayecarumba:
I get your point, but I think Marilyn's table was set up like this:

Behind door 1 Behind door 2 Behind door 3 Result if staying at door #1 Result if switching to the door offered
Goat Goat Car Wins goat Wins car
Goat Car Goat Wins goat Wins car
Car Goat Goat Wins car Wins goat


From this table the contestant wins the car once it he stays and twice if he switches. Also, Marilyn considers the goats to be indistinguishable, so there is no goat1 and goat2.

There are many solutions to this problem. Consider this:

You select a door. Monty gives you the choice of staying with your selection or opening both of the other doors and taking your choice. Isn't that the same proposition as showing you which of the other doors doesn't hide the car and letting you choose the other?
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Wizard
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Wizard
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January 17th, 2019 at 3:25:19 PM permalink
Quote: netzer

You select a door. Monty gives you the choice of staying with your selection or opening both of the other doors and taking your choice. Isn't that the same proposition as showing you which of the other doors doesn't hide the car and letting you choose the other?



Yes, it is.

As to DJ's point, I agree, this shouldn't be called the Monty Hall problem, because it doesn't reflect his behavior on the show. Yes, it was curtains, not doors.
It's not whether you win or lose; it's whether or not you had a good bet.
Ayecarumba
Ayecarumba
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January 17th, 2019 at 4:35:04 PM permalink
Quote: netzer

Ayecarumba:
I get your point, but I think Marilyn's table was set up like this:

Behind door 1 Behind door 2 Behind door 3 Result if staying at door #1 Result if switching to the door offered
Goat Goat Car Wins goat Wins car
Goat Car Goat Wins goat Wins car
Car Goat Goat Wins car Wins goat


From this table the contestant wins the car once it he stays and twice if he switches. Also, Marilyn considers the goats to be indistinguishable, so there is no goat1 and goat2.

There are many solutions to this problem. Consider this:

You select a door. Monty gives you the choice of staying with your selection or opening both of the other doors and taking your choice. Isn't that the same proposition as showing you which of the other doors doesn't hide the car and letting you choose the other?



Although none of the variables changed (no prizes switched locations), the odds of the initial selection begin correct changed due to Monty sharing information.

I hate to bring this example up, but how is this different than the "Two Dice" problem? (A game where the dealer shakes two dice under a bowl, then peeks under it and tells you, "one of the dice is a "2". What are the odds the other die is also a "2"?

It has been exhaustively explained that the odds remain 1/11, even with the "new" information that one of the dice is a "2". How does the "new" information in the Monty Hall problem change the odds, while the information in the "Two Dice" problem doesn't?
Simplicity is the ultimate sophistication - Leonardo da Vinci
netzer
netzer
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January 17th, 2019 at 5:18:28 PM permalink
Quote: Wizard

I agree, this shouldn't be called the Monty Hall problem, because it doesn't reflect his behavior on the show. Yes, it was curtains, not doors.


Whether we like it or not, the name seems to have stuck. It is essentially the same as Martin Gardner's Three Prisoners Problem and is closely related to Bertrand's Box Problem from a century earlier.

Bed time now in South Africa. Good night!
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netzer
netzer
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January 19th, 2019 at 8:55:40 AM permalink
Quote: Wizard


As to DJ's point, I agree, this shouldn't be called the Monty Hall problem, because it doesn't reflect his behavior on the show.


I think this is probably true. I have scanned a number of episodes of the show. He had a many games involving three doors with all sorts of merchandise behind them but I was unable to find a single instance involving two goats and a car. If anyone can find one, please share.

I did find an interview with Monty in which he said that he offered the contestant money instead of a switch.
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