Wednesday, January 6, 2010

The Bottom Line

Danimal said...

Ok I'll bite ... Happy New Year, by the way!

First I'd ask the prize-giver -- let's call him Pat Sajak -- whether there's a rule about how far the arrow has to go to be a valid spin. (I guess I'm conditioned by watching Vanna White, err, I mean watching Wheel of Fortune, as a kid.) It seems there should be some rule -- probably going around once or some number n times -- since otherwise I could just nudge the spinner a tiny bit and collect the $5. But if I were told there's no such rule, I'd nudge it and collect the$5!

Next I'd ask if I could have a practice spin. Motivation here is to find out how hard the arrow is to spin. If the arrow has to go around some integer number of times and is sufficiently hard to spin that you can barely get it around that many times, then the fact that it starts in the *middle* of the red zone could reduce the expected winnings from red to less than $3 (the payoff from the blue zone). If in the practice spin I find that the arrow is easy to spin, then in the money round I'd pick red and spin it as fast as I could, eliminating the effect of the arrow starting in the middle of the red zone, and making all the zones equally likely. If it were very hard to spin, I'd spin it hard as I could to make a valid spin and pick blue. If I couldn't ask any questions or get any more rules -- if this were just some inscrutable machine -- then I think I try the "nudge spin" first. I'd take the risk that my spin would be disqualified, b/c the reward of$5 is high. If I was told the spin were invalid, I'd spin the arrow as hard as I could, since that is indicated whether the wheel is hard or easy to spin.

Interesting -- it seems that in no case should I spin the wheel at half-strength -- either very light or very hard. At least with my assumptions...

JasonZimba said...

Happy 2010, Danimal! Hope you guys had a great holiday.

BTW I like your analysis - and now that you've been so helpful, I can confess why this is here. For various reasons, I often find myself reading through state math standards documents, and at times in doing so, I have had a disappointing feeling that probability problems are not being taught as mathematical models of physical processes. So I'm glad to see a physicist showing here how that might be done! In particular, I note that you keyed on the stickiness of the grommet, or whatever it is that serves as an axle. (It might also be important to ask whether this spinner is mounted vertically, as it might appear to be in the image.)

So one point of this heavily underspecified math problem is simply to make it impossible for someone to view it as a math problem.

But I should also explain why the geometry of the colors is the way it is. Under what appear to be "standard spinner assumptions," students in school are to assume that each spin eliminates the effect of the arrow's initial position, as you phrased it. Yet strangely, the students are then sometimes told that under these assumptions, they should assess the relative likelihood of the different outcomes by considering ratios of *areas* for each color - not ratios of angles or arc lengths.

Finding that notion peculiar in the extreme, I made up a situation in which ratios of areas of colors are an unreasonable approach - so much so, that even if one were initially thinking about area ratios, one might quickly change one's thinking.

A friend suggested the title "The Bottom Line," because in this silly little picture he actually saw something of a metaphor for solving the complex problems of life and society. Here, in order to decide what to do - in order to decide where to place your bet - you must (and you will) collapse a chaotic jumble of colors and shapes down to a single determinant. My friend was intrigued by the power of such "willful reductionism" in revealing the right strategy.

Danimal said...

Jason, thanks for the clarification. I had a feeling that the purpose of the question might be a distinction between the colored area and the relevant "area" for a spin(the angular measure along the circumference). And I wondered if I was overcomplicating things with my analysis. So glad to hear it was helpful, and that that was part of your purpose -- to complicate the "standard spinner problem".

I'm kind of curious at what point you think it would be best for K-12 students to encounter a problem like this spinner question. Is there a role for a stage where they learn the rules with "standard spinner assumptions" and do standard exercises, and then only later (in a year or two, say) encounter problems where those rules/ assumptions are challenged? Or is it better to present some questions like this one, requiring more flexible and realistic thinking, from the start?

This makes me think of an Emily Dickinson poem, which seems to suggest the first option. It's short so I'll just cut and paste it here -- you probably have seen it before --

"Tell all the Truth but tell it slant---
Success in Cirrcuit lies
Too bright for our infirm Delight
The Truth's superb surprise
As Lightening to the Children eased
With explanation kind