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Agent 007phd, Licenced To Err


eel_shepherd
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A less than serious topic, but a cautionary one even so. Beware of the Argument From Authority.

 

Below are some quotations, almost all by current PhD's (...[P]iled [H]igher & [D]eeper...), and the rest from probable future PhDs, taken from Martin Gardner's book "Weird Water, Fuzzy Logic" or somesuch name. It deals with the august utterances from some august academics, given as reader feedback to a column by World's Smartest Person Marilyn vos Savant (anyone want to touch that? I'm not going near it...), on the Monte Hall Problem. For those who are unfamiliar with the MHP, it goes like this:

 

A guy is walking along the road and he meets a man standing in front of three [3] doors, who says to him, "Behind one of these doors is a Cadillac and behind each of the others is a goat. I'm gonna let you choose one of these doors, and whatever's behind it, you get to keep for free. And by the way, I _know_ what's behind each door." The guy then selects one of the doors, and is getting ready to find out what's behind it, when Monte says, "Wait a sec; before you look behind that door, I'm gonna show you what's behind one of the doors you _didn't_ pick." And then opens one of the other two doors, revealing....... a goat. He then goes on to say, "Now, I'm gonna give you a chance to either stick with the door you originally chose, or switch your choice to the remaining closed door." What should the guy do? (Always assuming he doesn't actually prefer the goat to the Cadillac...)

 

Well, call me a spoilsport, but I'm gonna give away the answer, which is that the person should _switch_ to the remaining closed door, abandoning his original choice. The Cadillac is twice as likely to be behind the third door as it is to be behind the guy's original door.

 

Bridge players will recognise this scenario as being identical in kind to the situation of having a 9-card trump suit, missing the Q-J-x-x, with K-10-x-x in the Dummy and A-9-x-x-x in the hand, and trying to pick up the opposing trumps without losing any tricks in it, in the situation where you first lead the Ace and the fourth player gratuitously drops the Queen (or Jack) under it. When you next lead towards the K-10-x, and the 2nd player plays a small card, what do you do? Play the Ten and finesse him out of his Jack (or Queen), or play the Ace, hoping to crash the doubleton Q-J from the fourth hand? It's called The Principle Of Restricted Choice. Bridge writer Terence Reese said that this is one of those situations that some people just don't "get", while to other people the matter is so straightforward that they can barely believe that there is anything _to_ get. It seems to be one of those idea-shapes that come hardwired with some people and not with others. (I confess, I belonged to the first group, and when I described it to my girlfriend at the time, it was clear that she was in the latter group, as she was obviously waiting for me to get to the "problem" part of the question.)

 

Anyway, long set-up. Here's the PhDs reader feedback to Marilyn, after discovering (or failing to discover....) that they evidently did _not_ come hardwired with it (the parenthetical comments are mine):

 

 

"I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error..."

- Robert Sachs, Ph.D., George Mason University ("You're a witch, aren't you? Confess!!!")

 

"You blew it, and you blew it big!... You seem to have difficulty grasping the basic principle at work here.... There is enough mathematical illiteracy in this country, and we don't need the world's highest I.Q. propagating more. Shame!"

- Scott Smith, Ph.D., (currently grasping and blowing it big at...) University of Florida

 

"Your answer to the question is in error. But if it is any consolation, many of my colleagues have also been stumped by this problem."

- Barry Pasternack, Ph.D., (no, Barry, not many of YOUR colleagues --- one of THEIRS...) California Faculty Association

 

 

A SECOND, explanatory, article by Marilyn vos Savant evidently had no effect on the following die-hards:

 

"I am in shock that after being corrected by at least three mathematicians, you still do not see your mistake."

- Kent Ford, Dickinson State University (not a Ph.D., and, hopefully, now out of the emergency ward)

 

"...Albert Einstein [...name dropper... - ed.] earned a dearer place in the hearts of the people after he admitted his errors."

- Frank Rose, Ph.D., University of Michigan

 

"...Your answer is clearly at odds with the truth."

- James Rauff, Ph.D., Millikin University (...define "clearly", Jimmy...)

 

"May I suggest that you obtain and refer to a standard textbook on probability..."

- Charles Reid, Ph.D., (and, we suspect, author of probability textbooks) University of Florida

 

"...I am sure you will receive many letters from high school and college students. Perhaps you should keep a few addresses for help with future columns."

- W. Robert Smith (a likely story...), Ph.D., Georgia State University, (currently making a useful list of new pen-pals)

 

"You are utterly incorrect... How many irate mathematicians are needed to get you to change your mind?"

- E. Ray Bobo, Ph.D., Georgetown University, (currently working on the "how many irate Ph.D.s does it take to change a lightbulb" problem)

 

"...If all those Ph.D.s were wrong, the country would be in very serious trouble."

- Everett Harman, Ph.D., U.S. Army Research Institute (this guy's not content to just be wrong; he's also meta-wrong)

 

"Maybe women look at math problems differently than men."

- Don Edwards, Sunriver, Oregon (...only some women and some men, Don...)

 

 

==================================================

 

Once again, beware the Argument From Authority. As chess grandmaster Savily Tartakover said, "It's not enough to be a good player; you must also play well."

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<snip>Well, call me a spoilsport, but I'm gonna give away the answer, which is that the person should _switch_ to the remaining closed door, abandoning his original choice. The Cadillac is twice as likely to be behind the third door as it is to be behind the guy's original door.

 

<snip>Bridge writer Terence Reese said that this is one of those situations that some people just don't "get", while to other people the matter is so straightforward that they can barely believe that there is anything _to_ get. It seems to be one of those idea-shapes that come hardwired with some people and not with others. (I confess, I belonged to the first group, and when I described it to my girlfriend at the time, it was clear that she was in the latter group, as she was obviously waiting for me to get to the "problem" part of the question.)

Eel Shepherd, I'm in the group with your girlfriend. :shrug:

 

Initially each choice has a 1 in 3 chance of being the door with the Cadillac, right?

 

He picks a door, then it is revealed a goat is behind one of the other doors.

 

So now there are two closed doors left.

 

It seems the two doors left would each have a 1 out of 2 odds of being the one with the Cadillac, no?

 

Why is it to be twice as likely to be in the door he had not chosen originally?

 

Someone please tell me! :beg:

 

BTW, I don't play Bridge either.

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My son just explained this to me! Kuel!

 

The answer is also here

 

That site helped me understand it too.

 

I can't believe all those professors, once shown the table on the site I referenced, would still claim she was wrong. :shrug:

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Basically, someone else coming in fresh and picking from the 2 doors would have a 50/50 chance. Thats not whats happening here though. He picked a door out of three. He had a 1/3 chance of getting the right door. It's debatable, but there is a two in three chance that the car is behind one of the other two. Whether one is revealed to be a goat doesn't change that. It's a little difficult to think around. The point is there is a 2/3 chance that the car is not behind the first door he chose, so 2/3 times it will not be behind his door. Period. when you remove one of the other doors, you don't move the car. The car is still not behind his door 2/3 of the time. Remember that the removed door is not done by pure chance and these are not independant events.

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