Psychologists Don't Know Math

stupefaction writes "The New York Times reports that an economist has exposed a mathematical fallacy at the heart of the experimental backing for the psychological theory of cognitive dissonance. The mistake is the same one that mathematicians both amateur and professional have made over the Monty Hall problem. From the article: "Like Monty Hall's choice of which door to open to reveal a goat, the monkey's choice of red over blue discloses information that changes the odds." The reporter John Tierney invites readers to comment on the goats-and-car paradox as well as on three other probabilistic brain-teasers."

Comments

Nice try!

[geekoid]

Like I'm going to click on a link with the word 'goat' in it.

Re:Nice try!

[Ilan Volow]

It's never a good sign when the words "reveal" and "Monty" are in the same sentence.

Hmmm....

[Otter]

1) " Psychologists Don't Know Math" is a rather inflammatory, inaccurate, braindead headline, even by local standards.

2) The issue seems easy enough to settle empirically, given a few monkeys and a bag of M&Ms, besides the fact that it seems to have been empirically settled decades ago anyway.

3) This is, though, a good opportunity to ridicule "21" for completely botching the Monty Hall problem, along with pretty much everything else relating to math, gambling and Boston-area geography.

Re:Hmmm....

[Gat0r30y]

2) The issue seems easy enough to settle empirically, given a few monkeys and a bag of M&Ms, besides the fact that it seems to have been empirically settled decades ago anyway. One would think, but as it turns out, there are too many complexities. You see, you have to consider the socio-economic background of the monkeys, their upbringing, and their inherent biases to figure out if they like green, blue or red M&M's best. You see, the monkeys have an inherent bias toward green, but only if they have been captured from the wild (where presumably green would be comforting, the color of trees and whatnot). And of course there is the political bias associated with red and blue, so it depends on whether the monkey's political biases. These are especially hard to sort out as monkeys tend to just throw feces at the other side, at every opportunity, so you can easily separate the two groups, but rarely can you tell which is which. Its difficult to determine if they like to eat blue M&M's because they themselves are blue (or feel blue, as depressed monkeys have a significant bias toward the blue M&M's) or because they are red as it were, and feel like eating the blue ones to get back at the other side.

Re:Hmmm....

[fractoid]

So the expected value of switching envelopes is 50% (0.5X + 2X), or 1.25X. This is wrong. If one envelope contains X and the other contains 2X then the expected gain G from switching is:
G = 50% * (Gained if we were holding X) + 50% * (Gained if we were holding 2X)
= 0.5 * (2X - X) + 0.5 * (X - 2X)
= 0

So switching envelopes doesn't change the expected value.

To be fair, mathemeticians didn't know math either

[ZombieRoboNinja]

Marilyn vos Savant explained the problem in Parade magazine, and a whole bunch of math professors wrote in to tell her that she was wrong... turns out it's kind of a bad idea to play "gotcha" with someone who has an IQ of 228.

Re:To be fair, mathemeticians didn't know math eit

[wurp]

I read one of Marilyn Vos Savant's books, and in it she listed 9 as a prime...

She does seem to be brilliant, but everyone makes mistakes, and calling them on them will educate them if they were wrong, and educate you otherwise.

Re:Ummm, I don't get it.

[Jeremy Erwin]

It's quite simple.

Suppose the car is behind door number one.

If you pick door number one, then Monty has a choice of picking door number two, or three. If you switch, you lose.

If you pick door number two, then Monty must open door number three. If you switch, you win.

If you pick door number three, then Monty must open door number two. If you switch, you win.

Monty's choice of which door to open is constrained in two out of three choices. Pick the door he didn't open, and you'll win two out of three times.

But the problem assumes that Monty has to offer you that choice. On the game show, he didn't.

Indeed

[sustik]

This reminds me the story my high school teacher told me:

Some researchers involved in pchycology (social behaviour etc.) came to high schools and drew up the friendship graph of the class. (Maybe school works differently where you live, we had a class of size 30-40 students attending exactly the same lectures.)

They assumed friendship to be mutual (if not, than it was not considered friendship). One clever cookie made the observation that almost always there is a group of 6 students who all friends to each other (a clique), or alternatively a group of 4 students, who do not like each other.

There were excited discussions among the researchers what social forces are the reason that one of the above situations always seemed to occur.

They were somewhat disillusioned when our math teacher explained them Ramsey's theorem. Since R(6, 4) is between 35 and 41, indeed one can expect either a frienship or hateship clique to appear with quite high probability... (This does not mean that properties of the frienship graph worth not examining, but one needs to know the math to do it properly.)

Pot, Kettle, Black

[ryu1232]

I started questioning this article before the end of the first sentence. An Economist, calling a Psychologist "wrong" about math?
One should remember what happens when you put 50 economists in a room - you get 100 opinions - one for each hand.
I recognize that the author of the article may be correct, I just couldn't help commenting on the first sentence.

You know who can't do math?

[geekoid]

HR people.

If you are sick on a Friday or Monday, they assume you are 'taking a long weekend' even though there is a 2/5 chance someone will be sick on those work days. 40% of the time it would be Monday or Friday. More so for a 4 day work week.

Re:The problem is a fallacy

[1729]

You only ever had two options ... one with a goat and one with a car. Thus your chance of picking the door with the car are 1/2... This is analogous to observing that a lottery ticket can either be a winning ticket or a losing ticket, and then concluding that the odds of winning the lottery are 1 in 2.

Cognitive Dissonance

[jdbolick]

Amusingly, cognitive dissonance theory predicts that psychologists will rationalize their error and insist that it doesn't invalidate their conclusions.

Put it into more physical/visual terms

[davidpfarrell]

My wife and step-son asked me to clarify this probability after getting home from watching "21".

I realized that the door analogy wasn't working as it didn't help them visualize 'possession of the odds'

Instead I explained it as follows:

We're going to play the game with 10 boxes - 9 boxes are empty and 1 box contains a prize.

My wife is asked to pick a box and she is handed the box that she chose.

Then my step-son is handed the other 9 boxes.

I then ask both my wife and step-son what each ones odds are of having the prize is. The agree on :

Wife : 1 in 10 (or 10%) chance of having the prize
Step-Son : 9 in 10 (or 90%) chance of having the prize

At this point I explain the physical-ness of my son 'holding the odds' - It is clear to both that he is in possession of 90% of the odds.

I ask my wife, at this moment, with her holding 1 box and he holding 9 boxes, if she would like to switch possession and trade her 1 box for his 9

She of course says 'heck yeah!'

They both have an 'ahah!' moment and I don't really have to go any further, but I did for completeness.

I make a statement that my step-sons 90% is evenly distributed across the boxes he posses - currently 9 of them.

Now I start opening my step-sons boxes, one at a time - Boxes guaranteed NOT to contain the prize

After opening one of the 9 boxes, leaving my step-son with 8 boxes, I point out that he is still in possession of 90% of the odds, but now those odds are distributed between the 8 remaining boxes.

Then you remove one more box, along with explanation, and they see the pattern - The odds stay the same, and are still in my step-son's possession, but are continuously distributed among fewer boxes.

Finally both my wife and step-son are each holding one box.

I bring back the fact that my step-son is still in possession of 90% of the odds, but that entire 90% is wrapped up in that one single box.

With a final closing - that they were patient enough to listen to, since they asked me to explain after all - I point out to my wife that, since she was willing to trade 1 box for 9 boxes earlier, she must certainly be willing (if not eager) to trade her 1 box for my step-son's 1 box.

They really connected the dots pretty fast once I placed the prize in a box and had them each holding the boxes - Putting a physical location to the odds.

Inaccurate?

[jpfed]

As someone who majored in psychology, worked in two labs, and read countless psychology papers, I can tell you that 99% of psychologists avoid math when possible, and the other 10% try to use it but make obvious errors.

To the psychology researcher, it's more about getting the "story" right than actually quantifying anything.

Re:To be fair, mathemeticians didn't know math eit

[STrinity]

Marilyn vos Savant explained the problem in Parade magazine, and a whole bunch of math professors wrote in to tell her that she was wrong

In that case the mathematicians were correct. Vos Savant left out a key criteria when explaining the problem -- that Monty Hall knew what was behind each door and always chose to open one containing the boobie prize. That gives the game a memory and gives the player an advantage in the second part. If Monty just chooses randomly, as Vos Savant's version implied, the mathematicians would be correct.

Re:To be fair, mathemeticians didn't know math eit

[Cajun Hell]

I read one of Marilyn Vos Savant's books, and in it she listed 9 as a prime...

But there's a more-than-50% chance that 9 is prime!

I test primeness by dividing the test-number by all integers, from 2 through the test-number's square root, looking for a zero remainder. So, first, I divided 9 by 2. I worked on this for a while, and ended up with a nonzero remainder. So far, 9 looks prime, and I've already tested half of the potential divisors! In fact, there's just one more potential divisor to try: the number 3. I'm almost done, and everything rides on this final calculation. There's a lot of uncertainty here.

What are the chances that 9 is just going to happen to be divisible by the very last potential divisor that I try? I'll grant you that the chances are non-zero; there really are some composite numbers out there. But the chances aren't one, either. For example, when I was testing 17 for primeness, the last potential divisor I tried was 4, and it didn't work. This last calculation could go either way.

So here we are, having tested half of the possible divisors, and so far 9 is looking prime and there's just one more divisor to test against. So, I ask you: do you want to bet 9's primeness/compositeness on this last calculation? I'll make it easier for you: I tell you right now, that 9 is just like 17, in that it is not divisible by 4. And then, I'll even give you an option: we can finish the calculation by dividing 9 by 3, or you can change your candidate divisor to 5, now that you know 4 doesn't work. Well.. what'll it be?

Re:They don't know math?

[The End Of Days]

You glossed over the part where most people don't know how to deal with problems that a psychologist is trained to handle. There's something to be said for the education.

After all, I have plenty of friends, and I'm in complete contact with my family, but they have no idea how to help me get through a bout of depression in anything approaching a concrete manner. Just being there isn't enough.

And I noticed further down where you market your experience with psychology. I'd just like to remind you, your personal evidence isn't any sort of justification for such sweeping statements.

I'd also like to remind you that your concept of "good people" seems a little skewed to me. I think you need to dwell a bit on how to remove so much of your personal bias from your opinions on general topics. You have no basis for positing that the world is shy of good people, because you only know a vanishingly small fraction of them.