Behind Monty Hall's Doors: Puzzle, Debate and Answer?

[Dbacksfan12]

Article I came across and thought it was interesting.  The article discusses the "Monty Hall problem", but also discusses the feedback Marilyn vos Savant got after her coverage of it.  Amusingly, the Times put this in the "Goats" collection.  

Warning:  The article is filled with advertisements.  

Link.

[Matt Ottinger]

The article (from more than twenty years ago) does eventually get around to making the simple point that the whole argument swings on:  Monty knows.

[chris319]

Suppose you're the producer of LMAD and you plan to play this deal once per week for the next 39 weeks. How many cars should you budget for and expect to give away?

[TLEberle]

Suppose you're the producer of LMAD and you plan to play this deal once per week for the next 39 weeks. How many cars should you budget for and expect to give away?

I budget for 20 and hope like hell that I only give away 13.

[Matt Ottinger]

Suppose you're the producer of LMAD and you plan to play this deal once per week for the next 39 weeks. How many cars should you budget for and expect to give away?

I budget for 20 and hope like hell that I only give away 13.

That might be a problem.  If the game is presented as we typically understand "The Monty Hall Problem", and if people do what they're supposed to do, the odds say you're going to lose 26 cars.  Now, some people aren't going to do what they're supposed to, and Monty might employ some of his patented smooth-talking to play with the odds a bit, but the basic, dry game is a 2/3 win probability when played correctly.

[TLEberle]

That might be a problem.  If the game is presented as we typically understand "The Monty Hall Problem", and if people do what they're supposed to do, the odds say you're going to lose 26 cars.  Now, some people aren't going to do what they're supposed to, and Monty might employ some of his patented smooth-talking to play with the odds a bit, but the basic, dry game is a 2/3 win probability when played correctly.

And here's where I become disappointed in myself. When the professor of the statistics/probability class did this, I demonstrated why people were falling for the fallacy, and just now I did it myself, reducing the problem to what happens after a door is opened, as opposed to taking into account the previous event that sets up the door being opened.

Budget for 26, hope that some of the contestantry don't get it, fall for the fast-talk or just are plain unlucky.

/Does knowing the way to get to the right answer but putting down something else count for partial credit?

[chris319]

You don't budget and hope, you budget. The correct answer is 26.

This is where knowing the odds of the problem has utility. The worst-case scenario (or best-case, depending on your point of view) is that the contestant will switch every time, theoretically resulting in 66.7% wins. In actual practice you might budget for 27 cars to allow for a little variance in the outcomes.

The classic Monty Hall problem does not provide for Monty offering inducements to switch or not switch.

[Dbacksfan12]

In actual practice you might budget for 27 cars to allow for a little variance in the outcomes

I'm not mathematically sound to figure it out, but to those that are, what would the standard deviation be?

[chris319]

what would the standard deviation be?

0.48

[BrandonFG]

Every time I read about the Monty Hall Problem, I come out of it with a headache*. Except for this time...it always fascinates me, and I always forget the "right" answer. Speaking of which, I forgot that back then it was still possible to get snarky with a columnist, just with snail mail instead of the online comments section. I'm bit surprised that college professors and Ph.D.'s would be such twats about it though**.

Does anyone know if Wayne's version uses the scenario, or is there any rhyme or reason to the deals there?

*/There's a reason I majored in journalism :-P
**//At least one apologized

[TLEberle]

I'm bit surprised that college professors and Ph.D.'s would be such twats about it though**.

Why would you be surprised? They're paid to hold that position of authority and they probably don't get much blowback from students or other faculty, so when they're told "erm, excuse me professor, but you in fact have that precisely backwards", it doesn't surprise me one bit.

/Next time this topic comes up I'll say that you have a 2/3rds chance of winning if you stick.

[chris319]

I'm going to blow the lid off the Monty Hall problem.

There is what I call "House Odds" outlined above. Given n trials and assuming the player switches every time, the odds are that the car will be won 66.7% of the time.

Then there are what I call "Player's Odds". To illustrate, suppose I have a fair die and give you six rolls of that die. What will be the distribution of the six rolls? Theoretically it will be

1 - 2 - 3 - 4 - 5 - 6

Now wipe the slate clean. I give you one roll of the die. What will the distribution of outcomes be? The answer is:

? ? ?

because the number of trials is less than the number of possible outcomes. On LMAD each player receives just one trial.

How's that headache, Brandon?

[Kevin Prather]

I'm going to blow the lid off the Monty Hall problem.
...
because the number of trials is less than the number of possible outcomes. On LMAD each player receives just one trial.

This was the same problem people had with Deal or No Deal. People tried to use pot odds to determine whether to deal or not. Doesn't always work.

[TLEberle]

Now wipe the slate clean. I give you one roll of the die. What will the distribution of outcomes be? The answer is:

? ? ?

because the number of trials is less than the number of possible outcomes. On LMAD each player receives just one trial.

But you can still apply knowledge of probability to know that your range is one through six, and that for a fair die you should get each number 1/6th of the time. Even though the die is rolled one time you still can know what to expect, if not the result.

LMAD has a game where you have a certain number of rolls to score 10 points. For the first prize you have four rolls, then three rolls, then two. Are you asserting that since a player has but one trial they should ignore the math required to compute the likelihood of acquiring the number of points needed in the rolls given and just fly by the seat of their pants?

[Matt Ottinger]

I'm going to blow the lid off the Monty Hall problem....
because the number of trials is less than the number of possible outcomes. On LMAD each player receives just one trial.

This was the same problem people had with Deal or No Deal. People tried to use pot odds to determine whether to deal or not. Doesn't always work.

Well, Deal or No Deal is a lot more complicated an exercise than the MHP.  The fact that the LMAD contestant gets one trial doesn't change anything.  The odds are what the odds are.  The only question is whether you want a better chance to win the car, or a worse chance to win the car.  You still might not win the car, but just because it's a single trial doesn't suddenly make it a 50/50 proposition.