In for a Penny

The weather beavers are foretelling precipitation – liquid dihydrogen oxide – today. Given the portents of drought that hover over the Yankee republic continually these days and offer some telling indication – possibly – of climate matters, I cannot complain. Besides, aside from a slight increase in the likelihood of motoring accident there is something about a moderate amount of rain that is soothing.

Speaking of likelihoods, one of my colleagues, Magnetic Inductance Force, pointed me to an article [Link] in Smithsonian that seems to be about just that. It relates some work by a Stanford U stats professor who looked at the randomness of coin tossing. This is, of course, the bread and butter of undergraduate statistics texts and courses since the coin, simplified to only having two sides, is the simplest of dice. The statement in stats texts of “assume a fair coin” is almost as common as my favorite “demonstration of this shall be left to the student to shew”, which reveals the English origin of the author if not an outlook on life. More later.

The article talks about departure from fairness – equal probability that each side of the coin will be displayed – for two mechanisms: flipping and spinning for an American penny. As a physicist this was less than engaging since calculation of the moments of inertia and mechanics of coins begins in sophomore classical mechanics. Or, at least, it did in my day. Point being is that almost all coins are asymmetric. Simply put, they are “fatter” on one side than the other. For the American penny the side with the portrait of POTUS #16 is the fat side.

What does this mean. Well, first we have to consider what flipping and spinning are. Flipping is the off-center application of a vertical pointing force to the coin so that it has a normal ballistic trajectory with the coin rotating about an axis across the coin and ideally perpendicular to gravity. Spinning is a rotation of the coin resting on a surface about an axis close to gravity direction.

For the flipped coin, we have a situation where, in the main, the rotation does not appreciably slow before the coin hits the “ground” and stops. Because the “fat” side spends more time facing up than down, there is a slight bias to that side being displayed.

The spun coin, on the other hand, does slow its rotation before it “grounds” and because the dip during nutation increases more when the “fat” side is down, the “skinny” side is preferentially displayed.

But the really good question is what makes this random? Film at 11?