# Coin Tossing: How Random Is It?

The coin toss has been used for dispute resolution for millennia, with the earliest known use dating back to the Roman Empire. Many decisions great and small have been made based on the outcome of an airborne metal disc, from corporate mergers to municipal matters to the naming of cities. Portland, Oregon would have been named Boston, Oregon if an 1845 coin flip had gone differently.

Today, flipped coins are used most prominently in professional sports to decide which team has first possession or the order of penalty kicks, or even to break ties. Occasionally, the outcome of a game hinges on the result of a coin toss.

In an infamous example, the 1998 Thanksgiving game between the Pittsburgh Steelers and Detroit Lions was decided on the result of a controversial overtime coin toss. Though the stadium and television audiences heard Steelers captain Jerome Bettis call "tails," referee Phil Luckett heard Bettis change his call in mid-word from "heads" to "tails" and went with the player's first choice. The coin came up tails, Luckett awarded the toss to the Lions, who took possession of the ball and went on to win 19-16.

The coin toss is often considered the epitome of randomness. The terms "flip of a coin" and "50-50 chance" are used interchangeably to describe the probability of an event with two equal outcomes.

To those with any understanding of physics, however, the flip of a coin should seem anything but random. A flipping coin must obey the laws of physics, and thus its outcome must also be determined by these laws. The outcome of a coin flip, it would follow, is based not on random chance but by the way it is flipped.

The idea certainly seems logical. Demonstrating it required a marriage of mathematics and modern technology.

## This Show Was Almost "Bostonia"

## The Physics of the Coin Flip

While the coin flip certainly seems random to most of us, for Persi Diaconis, professor of mathematics and statistics at Stanford University, the conventional wisdom just didn't add up. A trained magician who had paid his way through undergraduate school by performing magic tricks, Diaconis had taught himself through many hours of practice to flip a coin and have it land heads or tails at will. He knew that more than probability was at work - the coin flip was a physics problem with a mathematical solution.

Teaming up with fellow mathematics professors Joseph Keller, Diaconis began his project by commissioning some Harvard technicians to build a mechanical coin flipper. This spring-loaded device could launch a coin into a cushioned cup with startling precision - and with the same side up every time. This demonstrated conclusively that coin flips were based in physics rather than chance.

The next task was determining all of the motions at work on a flipping coin. For this, Diaconis and Keller were joined by University of California - Santa Cruz mathematics professor Richard Montgomery, who had previously developed the Falling Cat Theorem (seriously!) to explain how cats land on their feet.

The researchers proposed that there were a number of significant factors involved in determining the outcome of a coin toss: the amount of time the coin is airborne, the weight and thickness of the coin, the rate of rotation, and the coin's **precession** - the change in its axis of rotation. Based on their calculations, the researchers predicted that a tossed coin would have a slight bias to land the same way it started - a coin that started heads up would land heads up more than 50% of the time. Now they needed an experiment to find out just how much.

Enlisting the help of Stanford's electrical engineering department and their high-speed camera, Diaconis' team captured hundreds of frames of airborne coins. The task of analyzing the images to determine the exact motions of the coin in the air fell to Diaconis' wife, statistics associate professor Susan Holmes. Her careful analysis of the data confirmed the prediction - flipped coins were biased to land the same way they started 51% of the time.

## A Revolution in Numismatic Decision-Making

To say that the paper published by Diaconis' team rocked the world of probability would be an overstatement, though it did generate some cheekily-written news pieces at the time. The knowledge that coin flips are predictable does raise the question of what actually is random. The rolling of dice, though more complicated than a flipping coin, is nevertheless a similar physics problem. The same with spinning roulette wheels. The shuffling of cards is also essentially a physics problem with 52 motion vectors. Even the random-number generators in a computer chip aren't quite random, often using a long repeating decimal string to generate pseudo-random numbers.

For practical applications such as playing board games or dealing poker hands, knowledge of the physics involved won't make much of a difference. The roll of the dice in a game of craps is still, well, a crapshoot for all intents and purposes. However, if you do find yourself making a decision via coin flip, you would be well-advised to call the side of the coin that was up before the flip. The odds are 51-49 in your favor, give or take a few decimal points.

## Sources and Further Information

- The Not So Random Coin Toss : NPR

Flipping a coin may not be the fairest way to settle disputes. A team of mathematicians claims to have proven that if you start with a coin on your thumb, heads up, flip it and catch it in your hand, it's more likely to land heads up than tails. - Lifelong debunker takes on arbiter of neutral choices

Magician-turned-mathematician uncovers bias in a flip of a coin - Heads or tails?

It’s supposed to be so simple. Heads or tails? It’s been done countless times in a countless number of games. No problem. - RANDOM.ORG - Coin Flipper

This form allows you to flip virtual coins based on true randomness, which for many purposes is better than the pseudo-random number algorithms typically used in computer programs. - Dynamical Bias in the Coin Toss

P. Draconis, et al. We analyze the natural process of ﬂipping a coin which is caught in the hand. We prove that vigorously-ﬂipped coins are biased to come up the same way they started. - Gauge Theory of the Falling Cat

R. Montgomery. Kane and Scher [18] proposed a mechanical model in order to explain and better understand how a falling cat rights herself.

## Comments

Stanford...Stanford and Stanford, this university is amazing! I rarely read a scientific article where its name is not mentioned!!

Anyway, this was really an interesting article...I really believe that nothing is left to chance, any experiment or any phenomenon in anywhere is based on some laws and have some variables...it may be very difficult, almost impossible, to determine all these variables, but this does not make it absolutely random...it makes it random for us, but actually it is not!

And ,I also believe, that this is the main problem with psychology, because every single reaction we take is based on tons of variable since we were born..and it may be impossible to determine all these variables, and that's why the error in any psychological experiment or treatment is present with a large percentage, well..that was just a thought!

Couldn't we perform similar calculations for a die roll? Given the velocity, rotation, launch angle, and initial orientation of the die along with the relevant metrics of the table, couldn't we predict how the die will come to rest?

I sense another grant proposal!

Was the height the coin would rise equal to the distance back down? This would account for a mirrored path. Also, there is a bias of another sort, the difference in the amount of metal raised on each side.