[I originally wrote, "Scientific knowledge is the best available knowledge we have.", but, of course, that is not true. For example, the "best" knowledge may be that your mother (or other appropriate person) loves you—which is not scientific at all. So, rather than "best", I have to settle for most accurate.]
Most everyone has heard of the Monty Hall problem. If you haven't, you're in for a real treat. It is a simple guessing game derived from the "Let's Make a Deal" TV game show with host Monty Hall. There are three closed doors on stage and you are the contestant. Behind one door is a brand new car! Behind each of the others is a goat. After much overt consternation, much to the enjoyment of the crowd, you choose a door, say, door No. 1. Then, without opening your door, Monty opens one of the other two doors, say door No. 2, revealing a goat. (Monty always reveals a door with a goat behind it.) Now, Monty asks you if you want to switch your selection. Should you stay with door No. 1, your original choice, or should you switch doors to the remaining door No. 3? Or, does it make any difference?
This problem was originally posted to the American Statistician magazine in 1975 in a letter by Steve Selvin, but went viral (before the WWW) when Parade magazine columnist Marilyn vos Savant presented it in 1990.
After Marilyn gave her answer with a short explanation, she was inundated with cards and letters proclaiming her mistake. In a later column Marilyn writes, "I’m receiving thousands of letters, nearly all insisting that I’m wrong, including the Deputy Director of the Center for Defense Information and a Research Mathematical Statistician from the National Institutes of Health!" Marilyn, however, stuck with her answer offering more explanation.
Here are a few samples of letters:
Since you seem to enjoy coming straight to the point, I’ll do the same. You blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to 1/2. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and in the future being more careful.
Robert Sachs, Ph.D.
George Mason University
Robert Sachs, Ph.D.
George Mason University
You blew it, and you blew it big! Since you seem to have difficulty grasping the basic principle at work here, I’ll explain. After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your selection or not, the odds are the same. There is enough mathematical illiteracy in this country, and we don’t need the world’s highest IQ propagating more. Shame!
Scott Smith, Ph.D.
University of Florida
Scott Smith, Ph.D.
University of Florida
Your answer to the question is in error. But if it is any consolation, many of my academic colleagues have also been stumped by this problem.
Barry Pasternack, Ph.D.
California Faculty Association
Barry Pasternack, Ph.D.
California Faculty Association
Human nature has not changed in the past 20 years, and, just like responses on today's internet, the responses to Marilyn were not just declaring her error but declaring her the butt of error and ridicule.
Maybe women look at math problems differently than men.
Don Edwards
Sunriver, Oregon
Don Edwards
Sunriver, Oregon
You are the goat!
Glenn Calkins
Western State College
Glenn Calkins
Western State College
You made a mistake, but look at the positive side. If all those Ph.D.’s were wrong, the country would be in some very serious trouble.
Everett Harman, Ph.D.
U.S. Army Research Institute
Everett Harman, Ph.D.
U.S. Army Research Institute
The last one was more prescient than he realized. There are many lessons here. "Why Smart People are Stupid", "Failures of Kindness", "The Bliss of Ignorance", and Stuart Firestein's talk—all previous In Progress topics.
However, I wish to emphasize the lesson of science. Marilyn wrote that 92% of the general public logically concluded her answer was wrong, including 65% of the responses from universities. I don't know how many mathematicians (nearly 1,000 PhD's) argued against her solution, but there were many. One in particular was renown mathematician Paul Erdös. He was arguably the most prolific publisher of papers in mathematical history, "comparable only to Leonhard Euler" (Wikipedia), and one of the greatest mathematicians of this or any age. He died in 1998. If you wish to see a fascinating movie, watch "N is a Number: A Portrait of Paul Erdös". He, with all his logical and mathematical knowledge, disagreed with the answer Marilyn presented.
The only way Paul Erdös was convinced that the solution stated was correct was by seeing it tested. Do the experiment a hundred times or so and you will see that you should always switch doors, as Marilyn stated. Your chance of getting the car is double than if you don't switch—66% to 33%. Of course, goatherders may wish to take a different path.
____________
EDIT: Martin Gardner (who else) introduced a version of the problem (before the Monty Hall show) in his Scientific American column in 1959.
____________
EDIT: Martin Gardner (who else) introduced a version of the problem (before the Monty Hall show) in his Scientific American column in 1959.
6 comments:
I've got a working hypothesis -- not yet fully tested -- that says that all decent science at least at first is counterintuitive. The heliocentric theory of the solar system was counterintuitive. So was the theory of evolution, the germ theory of disease and the relativity of time. Quantum mechanics is mostly counterintuitive. String theory, if I ever came to understand it, no doubt would turn out to be counterintuitive. It's almost as if, if it's not counterintuitive, it's probably not good science. Niels Bohr said it this way to a young physicist: "Your theory is crazy, but it's not crazy enough to be true."
Indubitably.
First a quibble or two. I'm not sure "all decent science" is at first counterintuitive, but certainly most memorable science is. Secondly, I'd say quantum physics is completely counterintuitive, and that string theory is not science, at least as science is normally defined, and as I have defined it here.
But Myk presents the main point of the post: why is our reasoning or logic or intuition so counter? Thank goodness there is repeatable experience.
He's also correct when he said I was "ruminating" over "the task of science in the vast amount of uncertainty". This is the first regurgitation of those ruminations. Whether they become "tenacious posts" is still uncertain, but "Are You a Possibilian?" could just as well be the conclusion to my thoughts as an introduction.
But if I do continue, I'm currently leaning toward a rollicking romp through the forest of knowledge.
Never was a post more timely. Perhaps this series will stem the tide. See Welcome to the Age of Denial.
In support of the idea that science, while the best thing we have, is still a narrative, there is Friedrich Nietzsche:
“Parmenides said, “One cannot think of what is not.” We are at the other extreme and say, “What can be thought of must certainly be a fiction.”
"There are no facts, only interpretations."
Myk often captures my (intended) thoughts better than I do. I no longer think I will make this "a persistent post", but at one point I intended to reference this article by Jeffery S. Rosenthal which should delight anyone who enjoyed the Monty Hall problem. Towards the end of the piece he references Bayes' Theorem, which would then lead me into more thoughts about human thinking.
Bayes' Theorem has seen its popularity wax and wan over the past couple of centuries, but is extremely popular presently. If you really scrutinize this blog, you have seen it before. Peter H. posed a brainteaser which was correctly answered by Mike. It is not coincidence that both Pete and Mike study medicine where Bayes' Theorem is widely used in medical research.
Eventually I put the nix on this considered thread. Who wants to read about epistemology? To make it interesting and humorous would take more time and research than I have.
Eventually I would have lead back to science since the point of all this is that we still need to reason about the experimental results. Science is really (at least) two things. First, it is recording repeatable, testable experience, and second, it is making sense of them. We call these theorems and laws and principles.
And as Parmenides, Kant and hundreds of others have said, we must group data in ways we can understand. This isn't to downplay science. It still is the most accurate we have, but I can point out two recent examples of people failing to recognize that science is still a narrative.
Both involve species in biology. One was a scientific article which stated that researchers studying small organisms are finding the idea of species as not useful nor descriptive. The other was a post by Kottke which found it remarkable that all 7,500 varieties of apples (a species) are recognizable, but that the species, Brassica oleracea, had cultivars such as cabbage, kale, broccoli, brussels sprouts, and cauliflower. What Kottke failed to appreciate was that 'species' is a man-made, albeit scientific, interpretation.
Winston Churchill told us that "it has been said that democracy is the worst form of Government except all those other forms that have been tried..." Could we not say the same for science? It is the worst form of inquiry into the world, except for all the others that have been tried.
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