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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."
Don't tell the Scientologists... You'll only arm them!
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.
What I'm listening to now on Pandora...
From an older article by the same author article:
Since she gave her [correct] answer [to the Monty Hall Problem], Ms. vos Savant estimates she has received 10,000 letters, the great majority disagreeing with her. The most vehement criticism has come from mathematicians and scientists, who have alternated between gloating at her ("You are the goat!") and lamenting the nation's innumeracy.
Since some math PhDs got it wrong too, isn't it a bit disingenuous to claim its the psychologists are the issue as the article title states?
-- Political fascism requires a Fuhrer.
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.
No, changing your door choice changes your chances of winning from 1/3 to 2/3.
:-)
When you choose one door out of three, and one of those three was pre-chosen randomly to be "the winner", your chance of having picked the right door is 1/3. At least one of the other two doors is not the winner, so the fact that Monty can show you that one is not the winner doesn't change your chance of having chosen the winner.
HOWEVER, now your chance is the same (1/3), but the chance of either the door you chose or the remaining door closed door being the winner is 100%. Therefore the chance that the remaining door is the winner is 2/3. Switch doors to double your chances.
I have a BS in math (not statistically oriented, but I had the normal discrete math sequence) and I still had to think about this a lot before I switched answers from the wrong one to the right one
Wikipedia has a much better explanation. Basically, if you stick with your original door, you have a 1/3 chance of winning. If you switch, you have a 2/3 chance of winning: http://en.wikipedia.org/wiki/Monty_hall_problem#Solution
Try thinking of the monty hall problem with 1000 doors. Your initial pick of 1 door has 1/1000 of being correct. Monty then opens 998 of the other 999 doors to show that the prize is not there. Should you switch to the other remaining door when asked or not? (You should: the other door has probability 999/1000 of being the one with the prize) The thing you are missing in your analysis is the extra information gained when Monty opens the oher door.
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.
The Kruger Dunning explains most post on
You're missing something.
"It is NOT, because as Monty will always pick a door with a goat behind it, your choices are always going to be two"
Your argument *only* works if Monty opens a door *before* you pick. *And*, you get to pick *twice*. First time from three doors, second time from two doors.
You pick, from a choice of three, giving Monty a choice of two.
Your argument is based on the reverse, Monty being able to pick from three doors, and you only get two.
Do you see it now? You 'lock' a door, precluding Monty from choosing it.
Remember, since you have first pick, your chances of getting a goat are 2/3. Meaning you most likely picked a goat. Meaning when Monty reveals a goat, the remaining door is most likely a car.
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.
Cube On! (http://stores.ebay.com/PuzzleProz)
Speaking of skepticism, do you have any evidence for this Chernobyl MRI claim? I find the idea a bit ludicrous that swarms of truck-mounted MRIs were running around one of the most backward areas of Europe somehow scanning farm animals for radioactive contamination a mere nine years after the first MRI scan of a human body and only three years after the first commercial MRI installation in Europe. If you have sources to the contrary I'd be really curious to see them.
That's equivalent to providing a table with all possible outcomes of a roll of two dice (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12) and saying that they are all equally likely just because each outcome has one entry in the table, except what you have done is the logical inverse. The example of the dice is combining multiple outcomes and pretending they are one - you are taking one possibility and branching it on a variable that has no effect on your outcome: the door that Monty picks if you picked the car to start with. If you pick the car to begin with, the number of the door that Monty picks has no effect on your outcome. To be more precise, the number of the door that Monty picks NEVER affects your outcome. If you want to keep the Monty column, you should replace the numbers with the word GOAT and then get rid of all of the duplicate entries, and the table will then represent the probabilities correctly.
Pick #1, Monty opens #2 (switch = win)
Pick #2, Monty opens #1 (switch = win)
Pick #3, Monty opens #1 (switch = lose)
Pick #3, Monty opens #2 (switch = lose)
50/50 No, the four possibilities here are not equally likely. If the initial pick is random, then the probability that case 1 occurs is 1/3, the probability of case 2 is 1/3, and the probability that EITHER case 3 or case 4 occurs is 1/3.
If monkeys were unable to reliably distinguish red, blue and green M&Ms, then they would have no systematic preference for one colour over another, and the experiment would not have found statistically significant evidence for such a preference (whatever its cause). However the experiments did find the monkeys have preferences about which colours they like.
You could equally well run the experiment with three types of treat - say peanuts, brazil nuts and pecan nuts - as long as individual monkeys have preferences among them and not all monkeys have the same preference. Whether these preferences exist you can check beforehand with a simpler experiment, before you waste valuable monkey-hours on running the psychology one.
-- Ed Avis ed@membled.com
The problem with Blogs is that they are inevitabley the whining and yapping of dogs that don't know any better. The worthless opinion that you link to fails to explain the original experiment correctly before weighing in. While it doesn't add anything of value, I guess it lets you slur the reputation of Dr Chen which is what you apparently wanted to do.
Chen didn't try to prove that the experiment was definitely flawed - he showed that their own reasoning for why it was correct was not valid. That is there were no conclusions that could be derived from the experiment because their methodology was incorrect.
It's actually quite a nice point that he's made, although some other poster further up pointed out that there are much deeper flaws in the original experiment such as the complete lack of a decent control. If they'd rerun it with an animal that had monochrome vision and they still got a 2/3s result then they would have known that something was amiss.
As far as Chen's point goes, I suspect most, if not all animals with chromatic vision have a subtle ranking, because we have evolved to look for certain foods. This is suspected to be why humans can distinguish red/green combinations much better than others.
Slashdot: where don knuth is an idiot because he cant grasp the awesome power of php
Statistics are tricky and generally counter-intuitive. As my stats professor said, often the best mathematicians are among the worst statisticians.
Les Miserables Volume 1 now up with my reading of
Another way to look at the problem is to consider what you'd do if there were 99 doors, you picked one, then Monty opened 97 of the rest of the doors (leaving your door, and one other). Obviously in this case you'd switch.
The reason people don't switch may be related to regret theory. If you switch and lose, you'll feel really bad because it will feel like you just lost a car (even though you didn't technically have it to begin with). So people stick with their current choice.
You can register on Slashdot and put as many words in a row as you like, but that doesn't magically make the choice to switch "probability independent".
Remember: once Monty opens that goat door, you know for a fact that the door you've already picked is twice as likely to have a goat behind it than it is to have a car. That's 2:3 odds, regardless of how many doors are left. In a "probability independent" choice, you would not have this information.
Presumably one could use NMR (MRI) to look for certain isotopes produced in a reactor explosion such as Cesium-137 or tritium. Getting the spectra out would be very easy for tritium, but an absolute bitch for cesium (1/2 vs 3 1/2). But I am still not sure why you'd do it that way rather than using a much cheaper scintillation detector (for example).
I agree it would be interesting to have some links, so I hope GP isn't just talking out of his ass.
Obama likes poor people so much, he wants to make more of them.
It's like a trained reflex, when you have a link that starts with "goat" and also contains the letters "c" and "x"..
I think this was a pretty interesting psychology experiment to get people to click on the link. Even just reading the phrase out loud brings people to a halt!
There's no question that your story about a researcher with no clue what s/he was doing is repeated often in psychology, and probably in other fields as well.
However, your example from fMRI speaks to complete ignorance of the field, and I'd like to force you to defend it. Thousands of fMRI experiments have been carried out, and this standard for significance is often met. When you say "very unlikely to actually exist," I can't imagine what you're thinking, since this statement is so easily falsified (in fact, in many publicly available datasets). Although not everyone does power analysis well, it's no secret how large an effect can be detected with fMRI, as a function of a few dozen easily estimated variables. Detectable effects are in fact very common. Your example seems to assume Bonferroni correction, which is generally not optimal. But even if we assume that it is, many many experiments produce results that exceed these thresholds, even though they do mean you can't study subtle effects in small groups.
If you want to defend your view that fMRI users don't grasp that they don't grasp it (I'm sure I would be included in your blanket statement), then you are doing a profound disservice to science by not submitting your work to NeuroImage (or if you think you wouldn't get a fair review there, there are certainly other journals you should consider). I want to rephrase myself here, to be perfectly clear. If you really do know of a flaw in statistical practice that affects many thousands of studies and many millions (probably billions) of dollars of grant money, but that has escaped the notice of everyone in the field, and you haven't taken the time to submit your insight to a decent journal, then you are the worst kind of bad scientist.
Of course it's not secret that many researchers do the statistics poorly (even making allowances for what's practical), and it's inexcusable. It's often for the same reason you mentioned: people just do what they saw in some other article, without understanding it. And many such studies are underpowered, although that's a problematic issue in itself. I would also agree with the statement that the SPM approach to fMRI has some very worrisome weaknesses, although not the one you identified. However, I will say that if you want to make such sweeping claims about what is and isn't nonsense, in a field in which you have obviously only dabbled (if that), you should make the basis for your criticism clearer, as naive readers could easily get the wrong idea.
A mathematician should deduce the fact that Monty knows where the car is even if she is not told about it. An average person might not, but a mathematician should since it is part of her damned job!.