Jason Rosenhouse is Associate Professor of mathematics at James Madison University in Virginia and the author of The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser, which looks at one of the most interesting mathematical brain teasers of recent times. In the excerpt below Rosenhouse explains what it is like to be a professional mathematician and introduces The Monty Hall Problem.

Like all professional mathematicians, I take it for granted that most people will be bored and intimidated by what I do for a living. Math, after all, is the sole academic subject about which people brag of their ineptitude. “Oh,” says the typical well-meaning fellow making idle chitchat at some social gathering, “I was never any good at math.” Then he smiles sheepishly, secure in the knowledge that his innumeracy in some way reflects well on him. I have my world-weary stock answers to such statements. Usually I say, “Well, maybe you just never had the right teacher.” That defuses the situation nicely.

It is the rare person who fails to see humor in assigning to me the task of dividing up a check at a restaurant. You know, because I’m a mathematician. Like the elementary arithmetic used in check division is some sort of novelty act they train you for in graduate school. I used to reply with “Dividing up a check is applied math. I’m a pure mathematician,” but this elicits puzzled looks from those who thought mathematics was divided primarily into the cources they were forced to take in order to graduate versus the ones they could mercifully ignore. I find “Better have someone else do it. I’m not good with numbers” works pretty well.

I no longer grow vexed by those who ask, with perfect sincerity, how folks continue to do mathematical research when surely everything has been figured out by now. My patience is boundless for those who assure me that their grade-school nephew is quite the little math prodigy. When a student, after absorbing a scintillating presentation of, say, the mean-value theorem, asks me with disgust what it is good for, it does not even occur to me to grow annoyed. Instead I launch into a discourse about all of the practical benefits that accrue from an understanding of calculus. (“You know how when you flip a switch the lights come on? Ever wonder why that is? It’s because some really smart scientists like James Clerk Maxwell knew lots of calculus and figured out how to apply it to the problem of taming electricity. Kind of puts your whining into perspective, wouldn’t you say?”) And upon learning that a mainstream movie has a mathematical character, I feel cheated if that character and his profession are presented with any element of realism.

(Speaking of which, do you remember that 1966 Alfred Hitchcock movie *Torn Curtain*, the one where physicist Paul Newman goes to Leipzig in an attempt to elicit certain German military secrets? Remember the scene where Newman starts writing equations on the chalkboard, only to have an impatient East German scientist, disgusted by the primitive state of American physics, cut him off and finish the equations for him? Well, we don’t do that. We don’t finish each other’s equations. And that scene in *Good Will Hunting* where emotionally troubled math genius Matt Damon and Fields Medalist Stellan Skarsgard high-five each other after successfully performing some feat of elementary algebra? We don’t do that either. And don’t even get me started on Jeff Goldblum in *Jurassic Park *or Russell Crowe in *A Beautiful Mind*.)

I tolerate these things because for all the petty annoyances resulting from society’s impatience with math and science, being a mathematician has some considerable compensating advantages. My professional life is roughly equal parts doing mathematics and telling occasionally interested undergraduates about mathematics, which if you like math (and I *really* like math) is a fine professional life indeed. There is the pleasure of seeing the raised eyebrows on people’s faces when it dawns on them that since I am a mathematician I must have a PhD in the subject, which in turn means that I am very, very smart. And then there is the deference I am given when the conversation turns to topics of math and science (which it often does when I am in the room). That’s rather pleasant. Social conventions being what they are, it is quite rare that my opinion on number-related questions is challenged.

Unless, that is, we are discussing the Monty Hall problem.

In this little teaser we are asked to play the role of a game show contestant confronted with three identical doors. Behind one is a car; behind the other are two goats. The host of the show, referred to as Monty Hall, asks us to pick one of the doors. We chose a door but do not open it. Monty now opens a door different from our initial choice, careful always to open a door he knows to conceal a goat. We stipulate that if Monty has a choice of doors to open, then he chooses randomly from among his options. Month now gives us the options of either sticking with our original choice or switching to the one other unopened door. After making our decision, we win whatever is behind our door. Assuming that our goal is to maximize our chances of winning the car, what decision should we make?

…For now I will simply note that it takes a person of rare sangfroid to respond with patience and humility on being told that the correct answer is to switch doors. You can share with a college class the glories of the human intellect, the most beautiful theorems and sublime constructs ever to spring forth from three pounds of matter in a human skull, and they will dutifully jot it all down in their notes without a trace of passion. But tell them that you double your chances of winning by switching doors, and suddenly the swords are drawn and the temperature drops ten degrees.

I have tried on a number of occasions to try to understand the math behind the Monty Hall problem. Sadly, I have also failed on each of these to fully grok (now there’s a word that marks me) why switching doors is the better tactic. I suspect this just reinforces my belief that the human brain is fundamentally mathematically challenged, and that Mother Nature has never been able to instill an innate sense of numeracy. It’s why people still go to Vegas and think that they have a “system” to beat the odds (you can’t); why they can’t grasp how 4 billion years of evolution can go from microbe to man (it can); and why your chances of death from a terrorist attack is about the same as being killed in a lawn-mower accident (it is).

I guess that’s why I prefer words to numbers and why I’m more interested in the etymology of “Monty Hall Problem” than the statistical truths behind it.

I truly believe that we have reached the point where technology has become one with our world, and I am fairly confident when I say that we have passed the point of no return in our relationship with technology.I don’t mean this in a bad way, of course! Societal concerns aside… I just hope that as memory becomes less expensive, the possibility of downloading our memories onto a digital medium becomes a true reality. It’s a fantasy that I dream about every once in a while.(Submitted using Nintendo DS running [url=http://cid-2602f0e287041cef.spaces.live.com/blog/cns!2602F0E287041CEF!106.entry]R4i DS[/url] QDos)

If nayone here in the future is wondering how to truly grok Monty Hall, here’s one way. Repeat to yourself: “Monty opening another door won’t make me more likely to have chosen correctly in the first place. He can and will do that no matter what, so it doesn’t change anything.”

If 300 people play, each choosing one of three doors, how many would you expect to pick the car right off the bat? About 100, right? Ah, but what if I then tell you that after those 300 people chose a door but before their chosen door was opened, the host opened a remaining empty door? Will this tidbit somehow “go back in time” and make some 50 more people able to pick the car at the start? Of course not.

If you were playing a version of the game where you just pick a door and take whatever it hides, would you bribe the host with $20 to open another door before you open your door? I hope not. That way lies madness!