Before writing this, I divided 350,757 by 48. The result? Just over 7,307. Which means 48 people each tossed a coin 7307 times. Give that some meaning: If you sat down and tossed a coin once every second without pause, it would take you a little over two hours to toss it 7307 times.
Question is, would you want to do this? Probably not. But what if I tell you it’s in the pursuit of science? Then you might change your mind. Maybe not once every second for two hours without pause, but you might agree to perform 7,000 tosses. And that’s more or less what each of these 48 people did.
Question is, why did they do it? To answer that, we go back in time a decade-and-a-half.
In a 2007 paper, the mathematician Persi Diaconis—famed in mathematical circles for his skills in magic—and two colleagues reported a rather remarkable finding. “Vigorously flipped coins," they wrote, “tend to come up the same way they started.... For natural flips, the chance of coming up as started is about 0.51." ("Dynamical Bias in the Coin Toss", Persi Diaconis et al, SIAM Review, 2007, https://statweb.stanford.edu/~cgates/PERSI/papers/dyn_coin_07.pdf)
That is, if a coin is tails-up when flipped, it has a slightly higher chance of landing tails, rather than heads—0.51 to 0.49. Now this is so slightly higher that it makes no real difference on a single coin toss, like the one that starts off a tennis or cricket match. But instead, let’s say you have a bet with a friend that depends not on one, but a thousand tosses. Let’s say the bet is simply that when done, you will have called correctly more often than him. Let’s say you can peer closely to see which face is up before the tosser tosses, and you call that face, every time. In such an experiment, you’re likely to win your bet, because you can expect to call correctly about 510 times out of 1000.
This is what Diaconis and colleagues concluded. And why this slight preference for the starting position? They start by referring to a study that “showed that ... a vigorous flip, caught in the hand without bouncing, lands heads up half the time." But your garden variety coin toss is not usually so neat. “Naturally tossed coins obey the laws of mechanics," they explain, “and their flight is determined by their initial conditions." The coins also “precess": the coin’s rotation itself changes the nature of that rotation, as the coin flies through the air. This is just normal. Tops precess as they rotate. So does our planet Earth. This is why the North Pole points at the star Polaris today, but pointed at Alpha Draconis about 5,000 years ago, and will point at Vega in another 13,000 years.
In the case of the flipped coin, Diaconis and colleagues took into account its “angular momentum vector"—never mind what that means—and the angle the vector makes with the surface of the coin itself. I’m simplifying this somewhat here, but in short, they explain that if that angle is zero, there’s no precession. But a coin is almost never tossed that way.
If the angle is greater than 45°, the coin “wobbles around" but never turns over—when it is caught in the hand, it shows the face it started with. In fact, staying true to Diaconis’ roots as a magician, the paper notes that “magicians and gamblers can carry out such controlled flips which appear visually indistinguishable from normal flips." Meaning that they can control which way the coin lands. But less accomplished coin tossers, like me, cannot control that angle and thus the precession. So the coin lands unpredictably.
From there, the paper dives into plenty more exotic mathematics. But the researchers find that in coins flipped naturally, there’s enough precession “to force a bias of at least 0.01."
Going even further from there, Diaconis et al consider another way of using a coin for random decisions—spinning it rather than tossing it. This can result in “huge variations" from any expected 50-50 result, attributable to the shape of the coin’s edge and all that’s embossed on the coin. In an experiment at the University of California, Berkeley, students spun the US 1-cent coin, the “penny", 100 times each. These had a noticeable tendency to come up tails. In fact, some students reported more than 90% tails.
And if that’s surprising, think of how we usually toss a coin. It flies into the air and falls to the ground, sometimes spinning on its edge before coming to rest. Combine that spin with the effect of precession and we might have even more variation from 50-50 in how the coins come up.
Anyway: Diaconis’ bias is so slight that “to detect [it] would require 250,000 tosses. The classical assumptions of independence with probability 1/2 are pretty solid."
Naturally, this got scientists interested in testing the “counterintuitive prediction" that a flipped coin lands on the same side as it started with, 51% of the time. František Bartoš of the University of Amsterdam was one such. He got a group of 48 other researchers to agree to toss coins—46 different currencies and denominations—recording which face was up before they tossed, which was up when they caught the coin. Note that there is film footage of some of these intrepid scientists doing this work (e.g. https://www.youtube.com/watch?v=3xNg51mv-fk). In the service of science, after all.
Between the 48, they tossed coins a total of 350,757 times, “a number that—to the best of our knowledge—dwarfs all previous efforts." (“Fair Coins Tend to Land on the Same Side They Started: Evidence from 350,757 Flips", František Bartoš and 49 others, https://arxiv.org/pdf/2310.04153.pdf)
What was the result of all this diligent tossing?
First, note that their “data show no trace of a heads-tails bias. Specifically, we obtained 175,420 heads out of 350,757 tosses." Pretty much exactly 50%.
But remember, the tossers noted the coins’ start positions. This confirmed the Diaconis finding. “The coins landed how they started more often than 50%. Specifically, the data feature 178,078 same-side landings from 350,757 tosses." That’s 50.77%. Given the large number of tosses, that’s a significant difference from chance, or 50%. As Bartoš remarked, “If you bet a dollar on the outcome of a coin toss 1000 times, knowing the starting position of the coin toss would earn you US$19 on average."
Admittedly, not the best way to make a living. Still, please don’t let the tosser cover the coin as she tosses.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun.
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