Monty Hall

[PassedOut]

The NYT has a nice simulation of the Monty Hall problem, fun to try.

 

Goat or Car?

[Al_U_Card]

Counter-intuitive or just counter pig-headedness?

 

Whenever I second guess myself....you know what I mean...lol

 

OTOH, since the additional door reveal changes the odds....it is good to know that switching is like restricted choice; when you see one of the honors pop on the first round then hook the other guy!

[kenberg]

I am amazed at the longevity of this problem. It's been almost 20 years since someone came up to me while I was eating chili at Hamburger Hamlet ans asked about this problem. As maybe the entire world knows, Marylin vos Savant put this problem in Parade Magazine and a lot of people who should have known better wrote in to tell her that her solution was wrong. After the dust-up a friend, who I feel also should have known better, said that the people who wrote in should have run a computer simulation first. GIGO strikes. If you misunderstand the problem you are highly likely to misprogram the computer. And if you understand the problem correctly you don't need a computer.

 

 

My favorite offshoot of this: I was presenting this to a class of liberal arts students in their required math course. I gave some background by saying that Marilyn vos Savant was once in the Guinness book of World Records as having the highest recorded IQ, and that she had presented this problem in her column in Parade Magazine. A young female student promptly asked, with great incredulity: She has the world's highest IQ and she writes a column for Parade Magazine? Give that student an A.

 

Presenting this problem in a classroom setting is fun. I have them act it out with some students representing doors, others goats, and one a car. The person playing the host, who knows where the car is, quickly sees the point. As soon as the contestant chooses an unseen goat, the host realizes that the rules force him to show the other goat and thus give away the car if the contestant adopts the switch strategy.

[G_R__E_G]

I am amazed at the longevity of this problem. It's been almost 20 years since someone came up to me while I was eating chili at Hamburger Hamlet

 

Wow, I'm just amazed that it had such an impact on you that you still remember what you were eating and where. Mind you, a good bowl of chili can be quite memorable all on its own.

[helene_t]

Why's that amazing? Maybe it was a spectacular car, or a spectacular goat, or a spectacular host. In combination with a spectacular chili it sounds as worth remembering.

[G_R__E_G]

The OP did not mention the quality of the chili - that was only speculation on my part. For all I know it could have been a horrible bowl of chili, which could also make it memorable. Also, he was in a restaurant at the time so I'm sure there were no goats present - that would be a health code violation.

 

PS I'm attempting to remember what I had for dinner last night and I'm having no success.

[TimG]

The NYT has a nice simulation of the Monty Hall problem, fun to try.

 

Goat or Car?

What if you want a goat?

[goobers]

The NYT has a nice simulation of the Monty Hall problem, fun to try.

 

Goat or Car?

What if you want a goat?

Then obviously you should not switch B)

[goobers]

If you misunderstand the problem you are highly likely to misprogram the computer.

Unless you are doing something completely counter intuitive (ie having the computer change the doors or calculate things based on probability instead of just making 3 doors), your program will not go wrong, even if you truly thought it was 50-50.

 

Edit: By counter intuitive, I meant absurd.

[Al_U_Card]

Monty Hall, being Canadian, was pretty cagey. Didn't he mix it up by also offering the contents of a small box etc.?

[barmar]

Yes, if I were a contestant I would always take Carol Merrill's box. :)

[irdoz]

This is being given a run by the NY times because of this story...

 

http://www.nytimes.com/2008/04/08/science/...nyt&oref=slogin

 

It looks like some of the original key research on monkeys and cognitive dissonance was just restricted choice and not 'dissonance'

[helene_t]

PS I'm attempting to remember what I had for dinner last night and I'm having no success.

If you're starving, better go for the goat, at least u can eat it.

[matmat]

I like the yellow M&Ms. This makes me a bad monkey.

[kenberg]

The chili was excellent. I think I remember it all so clearly because it just was such a surprising experience to be enjoying my chili, far from campus, and suddenly have someone (a former student) come up and start asking questions about a column I had never read that discussed a show I had never watched. I also usually have trouble remembering what I ate last night.

 

Incidentally my first reaction (I also remember this!) was "Hey, this sounds like restricted choice".

 

 

Incidentally, I think that the real moral of the problem is this: If you want to draw conclusions based on observation, it is important to carefully state what it is that you observed. If the contestant chooses door 1 and Monty opens door 3 to show a goat, the observation is not "There is a goat behind door 3" but rather "Monty opened door 3 and showed a goat". Phrased this way, the MH problem emphasizes clear and accurate observation, a lesson applicable in many situations.

 

As to mis-programming the computer if you mis-understand the problem, I don't want to get into a long (and unwinnable) argument over this hypothetical example but I am sure we have all seen people do some pretty strange things while analyzing data on a computer. Maybe a person would get it right here. But he should get it right w/o the computer also, and many didn't. Give MH to a beginning programming class and see how it goes. I'm betting a fair number will screw it up.

[kenberg]

This is being given a run by the NY times because of this story...

 

http://www.nytimes.com/2008/04/08/science/...nyt&oref=slogin

 

It looks like some of the original key research on monkeys and cognitive dissonance was just restricted choice and not 'dissonance'

I'll be damned! Fascinating.

[Mbodell]

As to mis-programming the computer if you mis-understand the problem, I don't want to get into a long (and unwinnable) argument over this hypothetical example but I am sure we have all seen people do some pretty strange things while analyzing data on a computer. Maybe a person would get it right here. But he should get it right w/o the computer also, and many didn't. Give MH to a beginning programming class and see how it goes. I'm betting a fair number will screw it up.

Well if the problem is ambiguously stated you might program a computer like this:

 

choose a door for the prize randomly from 1 to 3

choose a door for the contestant to choose randomly from 1 to 3

choose a door for the host to open randomly from 1 to 3

 

now if host != prize and contestant != host:

if contestant == prize don't switch++

if contestant != prize switch++

else:

start again

 

Each time through the loop you'd get:

111, or 222, or 333 => retry

112, or 223, or 331 => ds++

113, or 221, or 332 => ds++

121, or 232, or 313 => retry

122, or 233, or 311 => retry

123, or 231, or 312 => s++

131, or 212, or 323 => retry

132, or 213, or 321 => s++

133, or 211, or 322 => retry

 

and you'd prove that it doesn't matter if you switch or not. That is if the host opens a door at random that could contain the prize then the information communicated isn't the same as if the host always chooses a door that doesn't have a prize. So if you program the wrong experiment you'll get the wrong result.

[goobers]

Maybe I just have a different definition of absurd.

 

Randomly place goats/car.

 

Randomly choose a door.

 

If door = car, win.

If door = goat, lose.

 

Loop.

 

PS I am not smart enough to understand your program :)

[luke warm]

i never did understand what the problem was in this... with 3 doors you have a 1 in 3 chance of getting the car... put another way, it's 2 out of 3 that you *won't* get the car... so if you're shown that one of the other doors holds a goat and given the choice to switch, you have to take it

[goobers]

The problem as presented is very counter intuitive. At the critical point, you have two doors remaining. Who wouldn't think it wasn't 50/50?

 

I was in 6th grade when I first heard this problem, and I had to draw a diagram to finally understand why it was not so.

[kenberg]

Maybe I just have a different definition of absurd.

 

Randomly place goats/car.

 

Randomly choose a door.

 

If door = car, win.

If door = goat, lose.

 

Loop.

 

PS I am not smart enough to understand your program :)

 

 

Oops. I just saw that you said: get a car, win. No The switcher loses if his first choice is a car, wins if hios first choice is a goat. That is, after Monty opens the door, the odds are 2 to 1 the car is behind the door that Monty hasn't opened and the guest did not, at first, choose. I'll leave the post that I wrote, assuming that you meant: The switcher wins whenever his first choice is a goat, thus 2/3 of the time.

 

 

Once you fet this far there is hardly a reason to run the program.

 

The problem was that people thought, wrongly, that this was equivalent to something like: Contestant is presented with three doors, host opens one to show a goat, now we ask which of the remaining two doors is the car behind.

 

Of course the above problem is not equivalent. But what I said before is that if a person does not understand the problem then he might well program the computer incorrectly. If you do understand the problem, as probably everyone reading this does, then there is no need to run the computer. I don't think anyone doubts that if you write a computer program that randomly designates 1,2,3 as a car, the other two as goats, and then have the computer randomly select one of the three numbers then the long term behavior will give the car 1/3 of the time.

 

looking back, it may be difficult to see how so many people, many with mathematical training, went astray. The fact is that they did. I myself have been known to say and do stupid things so they have my sympathy. At any rate, until these folks think the problem through, I suggest not letting them near a computer.

 

Note: When Restricted Choice was first put forth, I believe by Reese, there was also (so I understand) a great deal of controversy. Further, after it was accepted and maybe still today, it is not that unusual to see people applying it where it doesn't apply.

 

So maybe there is some subtlety in it.

[goobers]

Yes, I am aware of the original problem, and originally I was writing pseudocode to simulate the events. However, when you're thinking of programming out the chain of events, you realize that it can all be simplified into the above.