Monday, September 04, 2006

Flipping Idiot

It was one of those charming traditions of a bygone era. Whenever my brother and I would disagree on something relatively minor (who got shotgun privileges that day, for example), dad would reach into his pocket and pull out a quarter. Up it would go, then be caught, then be slapped firmly against his arm. Brother was always ‘heads,’ I was always ‘tails.’

Cool, huh? Except that I never won.

Well, during a recent discussion of the Monty Hall Problem, a friend at work said something about “fifty-fifty—like flipping a coin.” I couldn’t resist bringing up a study I ran across many years ago that said pennies, at least, tended toward one side by around a percent. (I have since diligently searched for information on that paper, but have so far been unable to find it again.)

“Have you ever heard of ‘The Pretender’?” I had not.
“Well, he’s a genius who was kidnapped by an evil corporation as a child to solve complex problems. Well, there’s one episode where he reads that if flip a nickel a certain way, it’ll land on ‘heads’ a certain percentage higher than tails, but he’s skeptical. He gets stuck in a motel room for two weeks waiting for someone, so he spends the entire time flipping this nickel and recording every result to find out. And in the end it turns out what he’d read was right. You remind me of him.”

Well, after a story like that, how can I not watch the show. “The Complete First Season” [1996] now sits on my DVD case. It includes, by the way, the coin flipping episode, though ten years caused my friends memory to be a little off on the particulars. That episode (“Curious Jarod,” show no. 4) ignited in me a curiosity over the old quarter-dollar decision-making no-win question. Why did I never win?

So a few days I ago I entered the vacated bedroom on my floor and transformed it into a statistical laboratory.

I modeled the setup on what I could remember of the Lincoln-cent-piece statistics. On the edge of a level, elevated surface the quarters were balanced so that when a force struck the center of the surface the coins would plummet toward the surface below, turning end over end on the way down. I augmented that recollected setup by selecting sixteen quarters of random age and wear (who’s to say both faces erode equally compared to their unsullied state?) and drawing a line dividing each one directly in half with a fine-point Sharpie.

In my thinking, half of the coins should start out their drops heads-up, half heads-down. Of the eight coins drawn for each plane, my dividing line was drawn vertically for two, horizontally for two, diagonally one way for two, and diagonally the other way for two. Each set of two was further divided in that for one an arrow was drawn to the right of the line and for the other, to the left. In this way, starting the coins out all with the arrows pointing outward and then switching to all arrows inward, their combined falls would approximate every possible flipping orientation.

My theory of a predominant side is that for it to come up more often on one face than the other, the coin itself must slow in its rotation when one side is facing more generally and the other side is facing more generally down, and that the rotation would then accelerate as those faces reverse attitudes. In this hypothetical case, one side will stare upward more often than the other, and thus land that side up more often as well.

And so I began testing and retesting my array of sixteen quarter-dollar pieces. (The fact that I did this mostly during the course of a single night, with barrages of metallic change repeatedly striking the platform below them, was greatly the chagrin of the neighbors down the hall.)

The results after a total of 768 (I had hoped to go at least to 1000, but time ran out due to my painstakingly precise process for setting up each drop) individual landings? Three hundred and sixty-five heads, four hundred and three tails! That’s 47.5 versus 52.5 percent!!! And if that favors ‘tails,’ ‘heads’ will be the more likely result of the old catch-and-invert-on-one’s-forearm routine.

Of course, with only sixteen actual individual coins used, the test is probably open to error…

Another good probability question is this: Since 1965, quarters have officially had a constitution of slightly under ninety-two percent copper, clad on each side with nickel, which makes up the remaining eight-and-a-third percent by weight. But in each real-world example, the copper sits more on one side of the coin than the other. By the table I’m looking at, Cu is 1.0036 times as dense as Ni. By how much will this affect the probabilities of any particular quarter?
Maybe someone could suggest to their highschooler as a science-fair-winning project. Be sure to drop me a link to the results!

As another aside: my preliminary results on another front would indicate that the quart-dollar’s propensity to fall heads-down are dramatically increased if the coin is spun on edge. Somewhere around fifty-five or fifty-six percent tails-up.

Anyway, even if my coin-flipping results are untainted by error, a 52.5% inclination toward one side doesn’t explain my virtually 100% losing streak. For that, I may have to fall back on that most unscientific of principles, luck.
(Well, that and the variety of personal flipping styles, perhaps?)

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