OT - Monty Hall resurfaces in Parade magazine

[QuikSand]

For those of you who have recollections (fond or otherwise) of our old favorite puzzle, the "Monty Hall conundrum," I thought I'd post this tidbit.

Recently, Marilyn vos Savant (author of a horribly unctious little column in Parade magazine, distriubuted with many Sunday newspapers) received an inquiry harkening back to the classic "Monty Hall" puzzle -- for which I have received my notoriety as a defender of the 2/3 puzzle answer.

Anyway... here's a followup article, including the letter that sparked the debate:

Ask Marilyn (http://www.parade.com/current/columns/askmarilyn.html)


As it turns out, I happen to agree with her 100% this time, though she is rather cavalierly overlooking the fact that the first time someone asked her the original Monty Hall question, she booted it... and had to come back a few weeks later (after getting tons of mail about it) to apologize and cover up her error with some weak semantics.

As I have said here before, I think this is the greatest puzzle in probability ever conceived, and it's amazing to me how many people still get it wrong and/or cling to their incorrect answer. The process of convincing people that the right answer is right is a fascinating one, in my book.

[Anrhydeddu]

I have always wondered if your beef with her had just been a matter of style and presentation, or if you believe her answers have been wrong (which I would find hard to believe).

[Abe Sargent]

I utilize empirical evidence in order to prove the probabilities of the Monty Hall problem.

Divide a piece of paper into three section labeled 1-2|3-4|5-6

Roll a die thus selecting a winner, but keep the result hidden. Have the person choose a door. Reveal an incorrect answer. Then allow them the option of bouncing but tell them that you want to run it 20 times with them staying. If the probability is truly 50/50, as they contest, then they should be right by staying 50% of the time. After you run the simulation about 3-5 times, without even needing to get to 20, they usually acquiesce because they see how it works. If not, you'll usually have between 4-8 "right" answers - somewhere around a third.

(If you get an statistically high result, like 10 or 11 out of 20, and they decided to keep going, explain statistical probabilty to them and offer to go to 50. If it's truly a 50/50 chance, they should have no problem going up, and then you can show an answer closer to 1/3 than 1/2)

-Anxiety

[QuikSand]

Originally posted by Anrhydeddu
I have always wondered if your beef with her had just been a matter of style and presentation, or if you believe her answers have been wrong (which I would find hard to believe).

The first time she answered the Monty Hall puzzle, she published that the resulting outcome was a 50/50 probability, and that it didn't matter whether you switched doors.

After being berated (on both sides) by countless people, she reevaluated the puzzle, and rescinded her original answer. She claimed that she was answering a different question than was intended based on the word structure... which is itself a forgivable sin.


My only real beef with her is the notion that her readers seem to hold and that she seems very comfotable perpetuating, that because she has a very high IQ, she is therefore an appropriate person to ask anything-- not only trivia or intelligence-dependant things, but personal matters and so forth. Why do people send her "should I dump my boyfriend" questions, I will never understand.

[kcchief19]

Well, I will help confound QS even further by making a confession. I consider myself a reasonably intelligent person, although I admit that statistical theory is far from my strength. Even so, I have never and likely will never understand the logic behind the 2/3 answer.

The easy part I get. The first time you guess your chance of winning is 1/3. I understand that 2/3 of the time you are going to be wrong. But after one choice is eliminated, I do not understand how the odds of that door being the right door goes from 1/3 to 2/3. The only thing that makes logic sense to my feeble mind is that the chance of the prize being behind the other door is still 1/3.

I'm a journalist by trade and I staggered through Stat 31 at Mizzou with a sketchy C, so I would never cite myself as a statistical authority. I'm shaky with the whole genre of standard deviations, margins of error and the like. So count me in the mass of ill-informed.

[ahbrady]

I have a degree in math, and we talked about this problem at length in one of my logic classes in college. The teacher went over it time and time again, but never convinced the entire class. I will admit that I held out for a while too.

QS, your problem with Marilyn is something I have stated many times to people. She's a very smart and logical person, but I never understood why people think that they should ask her political, personal, or religious types of questions. It also bothers me because I have no desire to read those questions. I enjoy the puzzles, get rid of the rest. Also, QS, I would consider you at least at her level of intelligence. Why does she get to write a column? I enjoy reading your puzzles more, although often I am clueless to the answers. Let's start a campaign for Quiksand to have a column.

[Anrhydeddu]

I agree fully about answering non-logical questions (even though some can be answered logically). I don't know if she can do the logical or math off of the top of her head but there is nothing to assume that she cannot, therefore that makes her much smarter than I. My beef of late is that there have been way too many typewriter type puzzles which are stupid, imo. But the fact that she is popular and communicating logic and math to the masses should be encouraged, imo.

[QuikSand]

Originally posted by Anrhydeddu
But the fact that she is popular and communicating logic and math to the masses should be encouraged, imo.

Though I have a distaste for her in the specific, I can't really refute that argument.

[SackAttack]

kcchief, you musta had the same stat professor at Mizzou that I did. The dude was brilliant during his lectures, but the text he used (which I think he wrote) was so utterly confusing that I never fully understood anything in that class. How I managed to not-fail is beyond me.

[MIJB#19]

even mathematically sophisticated folks can stumble on themThat's a relief.

[Passacaglia]

This is really interesting -- just before visiting this site, I was trying to remember what she had said originally (since I buy the 2/3 argument), and in my search, came across this site, that lists all the times she has been wrong about something.

www.wiskit.com/marilyn.html

Oddly enough, it didn't really mention what she said about the Monty Hall puzzle, but it was interesting nonetheless.

[QuikSand]

Passacaglia, the Monty Hall puzzle is the fourth one on the list of her errors from the site you linked to:

Direct Link to Monty Hall discussion (http://www.wiskit.com/marilyn/gameshow.html)

[QuikSand]

Incidentally, Marilyn's fall on the Monty Hall puzzle reinforces the importance of stating as part of the puzzle that "this is a hint we offer all our contestants..." or something of the sort -- to isolate the probability as the root of the puzzle, rather than the human psychology.

[Passacaglia]

So it is.

This is interesting -- that discussion makes it sound like Marilyn initially answered that the contestant should switch, but the initial 'criticism' was that the game show host would not run the game in such a manner. Personally, I've never thought the problem was supposed to simulate the game, but to simulate SOMETHING LIKE the game (that line sounds familiar), but this site doesn't include the actual question, so it is hard to say.

[QuikSand]

Actually, my recolection is that she initially posted the question/puzzle and responded by saying "it's a 50/50 proposition now" - exactly the mistake that most people make.

Then, when she ran a follow-up (after receiving lots of mail) she said that the best answer to the properly constructed question/puzzle was actualy that switching increases your chances to 2/3. She suggested that she had been thrown off not by the math, but rather by the way the puzzle was written-- that is (in her mind) left open the possibility that the revealed door had been selected at random (which i find to be quite a stretch, but fits wit her answer).

Actually, I think the logic of the guy on the "Marilyn Was Wrong" website is a little weak. I think Marilyn got the puzzle right when she finally acceded that with th proper construct, the answer is to switch to get a 2/3 chance. The critic on the site seems to suggest that the puzzle's answer is itself unknowable - without considering that a simple tightening of the language resolves that ambiguity completely, and completely within the spirit of the intended puzzle.

[Passacaglia]

Incidentally, QuikSand, you do realize that you are the Marilyn Vos Savant of this board, right?

[Airhog]

QS: maybe you can explain to me why you get a 2/3rd increase when you switch doors? Does he only open a door if you picked a goat or something?

I guess I am one of those idoits.

[Abe Sargent]

Originally posted by Airhog
QS: maybe you can explain to me why you get a 2/3rd increase when you switch doors? Does he only open a door if you picked a goat or something?

I guess I am one of those idoits.

It's because you are thinking that the two doors is a new problem, but it's not. It's merely new information for the same problem. No matter what door you chose, at least one door will have livestock. Showing you that door will not change the chance that the doior you initially picked is right. But it does shift all of the chance under a third door.

If you are confused, do exactly what I suggest in my earlier response. Take a piece of paper and draw lines in order to divide it into thirds. (To represent picking three doors.) Label each door 1-2|3-4|5-6

Then pick a door. Roll a die to give you a "correct" door or a prize winning door.

Get out a piece of paper and a pen. Count the total number of times that you run through the simulation and the number of times that you are right, ok?

So, to recap, you have a die, three doors, and a scorecard.

Now, do simulation #1. Pick a door. Roll a die. Whether you are right or wrong, doesn't matter. You know which door is the right one. Now, eliminate a door by putting your hand over it or something. You now have a choice to move or not. Choose not to move each time.

Now, record whether or not you selected the right door. End simulation #1.

Repeat. Each time, choose NOT TO MOVE, but stay with your initial choice after the revelation that a door is false. Do this 10, 20, 50 times. However, many you need.

If you are right, and its 50/50 post-revelation, then you should have chosen the right door 50% of the time, and you will have around 50% right answers. However, your results will show that you are, in fact, right only around a third of the time.

Just run the simulation - takes about 10 minutes. Then come back here and report what you found.

But, after explaining the scenario, maybe you already see what happens. Every time you select a door, you have a 1/3 chance of it being right. Simple revelation that one answer is wrong does not add new information regarding that fact. But, since your initial door has 1/3 chance of being right, it therefore has a 2/3 chance of being wrong. The only remaining door, then, has a 2/3 chance of being right, because it is the only remaining choice.


-Anxiety

[albionmoonlight]

Another way to look at it:

If you picked right door initially (which happens 1/3 of the time), then Monty will pick a door "at random" and there will be a goat behind it. You will lose if you switch.

If you picked the wrong door initially (which happens 2/3 of the time), then Monty will pick a door "with knowledge" and there will be a goat behind it. You will win if you switch.

If Monty made his decisions at random all of the time, then 1/2 of the time that you picked the wrong door, he would show you a car. However, he never does that because he makes that choice with knowledge of the game. That knowledge is what turns 1/2 into 2/3.

[QuikSand]

I love it when a meta-puzzle discussion turns into yet another discussion of the puzzle itself. Especially this particular puzzle. I love it, honestly.

[Airhog]

Originally posted by albionmoonlight
Another way to look at it:

If you picked right door initially (which happens 1/3 of the time), then Monty will pick a door "at random" and there will be a goat behind it. You will lose if you switch.

If you picked the wrong door initially (which happens 2/3 of the time), then Monty will pick a door "with knowledge" and there will be a goat behind it. You will win if you switch.

If Monty made his decisions at random all of the time, then 1/2 of the time that you picked the wrong door, he would show you a car. However, he never does that because he makes that choice with knowledge of the game. That knowledge is what turns 1/2 into 2/3.

thank you, that explains it in a way that makes perfect sense.

[cuervo72]

I still don't quite understand her reasoning towards applying the same approach to switching answers on multiple choice tests (ran a month or so ago maybe?). Was that assuming the student has no knowledge on the subject (if so, I don't think she stated that)? Wouldn't that be an unwise assumption?

[QuikSand]

Originally posted by cuervo72
I still don't quite understand her reasoning towards applying the same approach to switching answers on multiple choice tests (ran a month or so ago maybe?). Was that assuming the student has no knowledge on the subject (if so, I don't think she stated that)? Wouldn't that be an unwise assumption?

No, the difference there is that (if the setup is proper) it's very clear that the revelation of one answer being incorrect was done at random.

If the selection of the revealed wrong answer was done at random (and not with knowledge), then the two remaining choices are definitely 50/50. That is the essential difference between the two puzzles - it makes all the difference.

[cuervo72]

Originally posted by QuikSand
No, the difference there is that (if the setup is proper) it's very clear that the revelation of one answer being incorrect was done at random.

If the selection of the revealed wrong answer was done at random (and not with knowledge), then the two remaining choices are definitely 50/50. That is the essential difference between the two puzzles - it makes all the difference.

No, that part I understand. I guess I'm having trouble conveying clearly what my bone was with Marilyn regarding what she said about multiple choice answers....

My initial impression was about something she said at the end of a column while discussing the topic for the umpteenth time. I unfortunately can't recall exactly what she said, but she was probably implying that switching test answers didn't matter if the student had originally chosen at random - I don't remember her explicitly stating such. I'd say most good students don't make a practice of doing that though :)

[QuikSand]

Sorry, I don't read the column very often, and didn't see her original discussion. Can't offer much more on her comments, I'm afraid.