Coin tosses are a classic metaphor in economics for randomness. For instance, in his book about market efficiency, A Random Walk Down Wall Street , economist Burton Malkiel compares the price movements of the stock market to the random outcome of a flipped coin: "[S]ometimes one gets positive price changes for several days in a row; but sometimes when you are flipping a coin you also get a long string of ‘heads' in a row." According to Malkiel, mathematicians' terms for the sequences of numbers produced by any random process—in this case a coin flip—is known as a random walk. To him, this is exactly what stock price movements look like; hence the title of his book.
Similarly, Nassim Taleb, in Fooled by Randomness , points out that the seemingly amazing success of money managers at beating the market is often best explained by pure chance. Randomness alone could easily explain why any one manager could do well for several years in a row. Instead, people misperceive patterns in what are, in fact, purely random sequences, akin to the outcomes of a coin flip. And as everyone knows, coin flips produce "heads and tails with 50% odds each."
Lately, the idea of randomness in stock market prices has come under attack; prices for individual stocks (but not the market on the whole) often show small momentum effects: Stocks that go up tend to keep going up, and stocks that are going down tend to keep going down. But the metaphor of a coin flip for randomness remains unquestioned. We use coin tosses to settle disputes and decide outcomes because we believe they are unbiased with 50-50 odds.
Yet recent research into coin flips has discovered that the laws of mechanics determine the outcome of coin tosses: The startling finding is they aren't random. Instead, for natural flips, the chance of a coin coming up on the same side as it started is about 51 percent. Heads facing up predicts heads; tails facing up predicts tails.
Three academics—Persi Diaconis, Susan Holmes, and Richard Montgomery—through vigorous analysis made an interesting discovery at Stanford University. As they note in their published results, "Dynamical Bias in the Coin Toss ," laws of mechanics govern coin flips, meaning, "their flight is determined by their initial conditions."
The physics—and math—behind this discovery are very complex. But some of the basic ideas are simple: If the force of the flip is the same, the outcome is the same. To understand more about flips, the academics built a coin-tossing machine and filmed it using a slow-motion camera. This confirmed that the outcome of flips isn't random. The machine could make the toss come out heads every time.
When people, rather than a machine, flipped the coin, results were less predictable, but there was still a slight physical bias favoring the position the coin started in. If the coin started heads up, then it would land heads up 51 percent of the time. Part of the reason real flips are less certain isn't just that the force of the flip can vary; it's that coins flipped by humans tend to rotate around several axes at once. Flipped coins tumble over and over, but they also spin around and around, like pizza dough being twirled. This spinning around is technically known as "precession." The greater the precession in a flip, the more unpredictable the outcome.
I spoke to Holmes, a statistician at Stanford, about her research. She told me that when most people hear about this weird finding, they think it has something to do with the density of the coin, but she was able to disprove this by constructing a coin made out of balsa wood on one face and metal on the other. This made no difference to the flips. The dynamics of the coin flip, and its outcome, aren't determined by the lack of balance in the coin but instead by the physics of spinning and flipping.
The true laws of coin tosses show yet again the inadequacy of our intuition (as well as the flawed metaphors favored by economists). We are indeed fooled by randomness. But we are also fooled by nearly random processes that look random, even if they aren't, because the differences are too subtle for us to notice. And hence we continue to use coin flips as a figure of speech but also in real life, particularly in gambling and professional sports.
I asked Holmes if coin flips used for, say, football, should be eliminated because they are biased. She pointed out that there is no reason to change the way the coin flip is done, as long as the person calling the flip doesn't know how the coin is going to start out. In football, the tosser is never the caller; the tosser is supposed to be a referee. But if you are both the caller and the tosser ... well, that changes things. Knowing about the bias in coin tosses give you an edge, albeit a tiny one.
Holmes admits she still uses the metaphor of a coin toss in her statistics class at Stanford all the time—after all, the randomness in a coin toss is off only very, very slightly, with odds being 51-49. But certain people, when they flip a coin, can make it come out heads (or tails) 100 percent of the time. Diaconis, Holmes' co-author and husband, is one of the people with this amazing talent. Before becoming a mathematician, he was a professional magician. Among his proofs is that it requires a full seven shuffles to truly randomize a deck of cards . He was admitted to graduate school at Harvard University after two of his card tricks were published in Scientific American.
So how exactly is Diaconis able to make a coin toss come out a certain way? Susan Holmes won't tell me: "It comes from his previous career-it's magic."