Tuesday, April 11, 2006

The Factorial Monty

O.K, here be the disclaimer: If you don’t know me very well (particularly if you don’t know anything about math) and you have run across this posting, you might be tempted to categorize me as a ‘math geek.’ Please do not do this.
I do not ask you this because to do so would be to insult me, but because to do so would be to insult every math geek on the face of the earth.
From a practical standpoint, I hate math. No offense to you mathies out there, it’s just not my area. However, I do tend to enjoy (at least on an abstract level) the workings of the world. So, in the end, I can enjoy math as an idea, so long as I am not expected to sit down with paper and pen and actually do anything.
(There are, of course, many exceptions to this rule. But let us not get into that.)

Now that that is out of the way…

There is a mathematical diversion known popularly as the Monty Hall problem. I vaguely remember hearing something about it ten years ago or so, but it was really only introduced to me recently, upon reading this book.

The problem, named for the host of that televised outcropping of the sixties known as “Let’s Make a Deal,” tied my brain in knots for several hours of intermittent examination until I finally rectified truth with assumption. Or, more accurately, finally vanquished a superficial assumption.
The problem also happens to be very timely as NBC is running something called “Deal or No Deal” which appears to be an attempt at remaking Mr. Hall’s legendary show (minus all the amusing trappings, and thus the point of the show). I haven’t actually seen any of this new program, of course…

But various acquaintances have, and have spoken of it. Which led to a remodeling of Monty Hall’s perplexing problem. The problem, as I presented it to the group, was as follows:
You are on TV. I point your attention to three models standing on a stage with plastic smiles, each with a briefcase in hand. One briefcase holds a gold brick and a million dollars in cash. The other two contain rats. In a moment you will be asked to select one case, whose contents you would like to call your own in the hopes that it has the riches. After you have selected I will have one of the models open one of the two boxes not selected, revealing one of the rodents. Understood? Good. Now, pick.
By overwhelming majority, it was agreed that the center box would be chosen.
Right, then. Miss Maine, yes, the one on the right, would you kindly open your attaché case? Thank you.
A white rat is shown to the audience and Miss Maine leaves the stage with it.
Now, honored-and-hopefully-lucky contestant, I am going to make you an offer. You may either stay with your original choice of what had been the center case, or you may switch and take home the contents of the case to its left. What is your decision?
One person said that she might switch, but that she didn’t really care. Everyone else present was emphatic: They were sticking to the original choice. (The reason for this, as explained to me afterwards, was that to switch and lose would make the player feel stupid, whereas to stick and lose would entail less of a sting.)
All right, can anyone tell me what the odds are of the original choice being correct, and what the odds are of the switch you all turned down having the fortune?
Fifty-fifty, everyone said. All but our resident math genius, who claimed they should both still be consider a one-third chance, having originated from three choices.

So, was fifty-fifty the correct answer? No.
When I informed the group that they had just received a suitcase with a lab rat inside, I had the makings of a riotous mob on my hands. A few simply stalked off and the rest angrily accused me of rigging the game. According to them, whatever they had chosen I would have claimed I had placed the goodies in the other case.
No, I said, it was purely a matter of probabilities. Sticking to their guns as they did, they would only get the loot one in three tries. Had they switched when given the offer, their chance would have doubled to two in three.

I was accused of imbecilic math (as was Marilyn vos Savant for getting it right the first answer, something I fell far short of).

But yes, it is true. The way I finally explained it to myself after perhaps five hours of off-and-on obsessing was that the initial choice (the center box, in this case) had a one in three chance from the beginning. Intentionally taking away one of the bad boxes doesn’t change that, so by subtraction the other box must necessarily have a value of two-thirds.

It became even clearer in my mind when an engineer I mentioned it to that first day tried to convince my of my error by illustrating it with a hundred boxes instead of three. The point that was supposed to made through this exercise was easily defeated by it in the end.
If you stick with your initial choice as ninety-eight bad boxes are removed, your initial choice still has only a one over a hundred chance of being correct. And since the other boxes were removed specifically for their identities as dead fish the other one left has a ninety-nine percent chance of holding the golden egg.

These were the terms I explained it in (being sure to emphasize the fact that I had been wrong-headed about Monty Hall originally as well), and I think everyone more or less agreed with and forgave me at the end.

Assuming that anyone has waded through my laborious explanation of an otherwise pretty cool concept, the interesting part happened next.
A woman, whom I do not know well but with whom I am acquainted, was attracted to my general area by the sounds of our discussion. She brought with her a problem of her own:
If an individual has five drawers, how many different ways are there of arranging said drawers within their five spaces?

I should mention that this woman has more mathematical education in her upper spinal chord than I have in my entire nervous system, aside from being a very intrinsically smart cookie. That, and the fact that her question was inspired by the discussion of an apparent paradox might lead one to believe the posed question was meant as some kind of test.

But it’s a simple enough problem. Actually, so simple as to be too easy, the kind of semi-circular reasoning that leads right back the idea of a trick question. But the simple truth remains that, as any one of the five drawers can sit in the first slot, but only one of the other four can sit in the next (and so on) 5 times 4 times 3 times 2 (times 1—but who cares?).
I write the numbers in the air briefly and cough up the number 120.

“That’s what I thought,” she says. “But I wrote out the possibilities and couldn’t get more than twenty.” Hmm. “Is the factorial over something?”

No…

She suggests I puzzle it through. I stared into space a bit, wrinkling my mouth different directions. I was trying to figure out how she only came up with twenty.

During the course of my duties that day (volunteering with a children’s program), I happened to be in contact with her fifth-grade son, who saw my (by then completed) work on the matter and asked what it was. I told him his mother had asked me to prove how many ways five objects could be arranged. It was then revealed not to be some purely academic exercise meant to trip me up.
“Oh, I know where she got that. See, we had these five drawers last night and she was trying to figure out how they were supposed to go in…”
I rather enjoyed that. When he suggested I write my proof with seven factors instead of five (as there had originally been two more drawers) I enjoyed it less. For my proof was a hand written list of all one hundred and twenty possible combinations of the five disoriented drawers.
Having retrieved some waste paper from the secretarial pool’s trashcan, I made out the list (exactly as below). Five columns of twenty-four combinations of five unique items.

O.K, so not exactly as represented here. The order and organization is the same, but my handwriting is terrible and I had to scribble out three duplications (two in the third row and one in the fourth).
Nonetheless, I was quite proud of myself.

When I met the lady of the mathematical furniture at the day‘s end, I forked my handiwork over with this simple speech: “One hundred and twenty.”
“You did do it.” She was surprised I’d bothered thinking of it at all.

Of course, as any genuine math junkie can probably see from the groupings within my list, I couldn’t have done it so fast (about twenty minutes while dealing with the kids in my care) without having the number 120 already in mind.
But I refuse not to be impressed with myself.

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