Fair Coins, Flipping Outcomes, and the Subtle Bias of Physics: What Makes a True 50-50?

Andrei Mihai

We have all struggled to make decisions at some point. Whether it is going to one restaurant or another, or deciding who starts first in a game, it is sometimes tempting to reach for an arbiter of fate, and the faithful, impartial coin is always there.

But what if that coin is not as impartial or random as it seems? What if, as new research suggests, the outcome is ever so slightly biased? Just to be clear, this is not about the infinitesimal odds of a coin landing on its side. We are talking about the two sides (“heads” and “tails”).

The appeal of a coin toss is simple. Coins are easily available and are perceived as fair. Many of us have used coin tosses to decide something at some point, but it is surprising just how many significant things in history were decided on a coin toss.

“A coin flip was used to determine which of the Wright brothers would attempt the first flight in 1903; who would get the last plane seat for the tour of rock star Buddy Holly (which crashed and left no survivors) in 1959; the winner of the European Championship semi-final soccer match between Italy and the Soviet Union (an event which Italy went on to win) in 1968; which of two companies would be awarded a public project in Toronto in 2003; and to break the tie in local political elections in the Philippines in both 2004 and 2013,” write the authors of a study investigating how fair coins are.

The authors of the study, led by František Bartoš from the University of Amsterdam, were inspired by a mathematical model proposed by in 2007. The coin-tossing model counterintuitively found that coin flipping was not 50-50 and the coin was more likely to land on the side it started on.

What Is in a Coin Flip

Coin flips are governed by deterministic physics. Factors like initial position, force applied, angular momentum, and air resistance essentially decide how the coin will land. In practicality, however, an honest coin toss can’t account for all of that. So far, nothing new. But the 2007 model introduced the idea that precession – a wobble in a coin’s rotation – favors the coin staying aligned with its starting position.

The model predicted a same-side outcome probability of around 51%, but it was not clear whether the model played out as expected in practice.

Granted, there has been no shortage of attempts to measure coin tosses.

In the eighteenth century, famed mathematician and naturalist Georges-Louis Leclerc, Count de Buffon, carried out one of the first statistical studies in history. He tossed a coin 4,040 times, which resulted in 2,048 heads, very close to 50/50 odds (the actual heads percentage was approximately 50.69%). In the 20th century, mathematician John Kerrich also flipped a coin 10,000 times while interned in Nazi-occupied Denmark. He recorded 5,067 instances of heads, or 50.67%, also very close to equal odds.

But there is a problem: Neither of them recorded what side the coin started on. This is what František Bartoš and collaborators set out to address. That and making a larger dataset, of course.

They gathered 48 participants who contributed nearly 351,000 coin flips. Coins from 44 different currencies and denominations were used, ensuring the findings were not tied to specific coin designs. Participants flipped coins in controlled sequences, recording each starting and landing position. To maintain integrity, flips were videotaped and audited for accuracy.

The results confirmed the predictions of the Diaconis-Holmes-Montgomery (DHM) model: The starting position influences the outcome slightly but consistently. No significant bias toward heads or tails was observed, underscoring the coin’s symmetry in this aspect, but there was a bias toward the starting position.

“We collected 350,757 coin flips and found strong empirical confirmation of the counterintuitive and precise prediction from DHM model of human coin tossing: When people flip a coin, it tends to land on the same side as it started,” Bartoš and colleagues note.

Overall, they found that the coin lands on the same side roughly 50.8% of the time, a marginal but statistically significant deviation from the expected 50%, particularly for such a big dataset.

Interestingly, the study uncovered another layer of complexity. Some individuals exhibited stronger same-side biases than others, likely due to the dynamic of their coin toss. However, as participants flipped more coins, the bias tended to diminish, suggesting that with practice, people learn to flip coins more symmetrically, reducing wobble and associated bias. It took around 10,000 flips – equivalent to about 10 hours of practice – to reduce this bias.

How to Do a True 50/50

The bias towards the starting side is small but significant. If you’d bet on a coin toss 1000 times and knew the starting side, you’d win $16 on average (maybe even more if the coin flipper isn’t “trained”). That’s comparable to the advantage the house has in roulette or blackjack.

But this does not mean you cannot use coin tosses fairly. The only prerequisite, Bartoš says, is that neither party knows which side the coin starts on. If that can be ensured, our coin toss is good to go. Please note that dice are also probably not truly fair, so they are likely not a good 50-50 generator).

Can You Use Random Number Generators (RNGs) Instead of Coins?

If the physical coin is biased, then perhaps a “digital” coin may be better.

Software-based RNGs, like those in programming libraries, typically use deterministic algorithms called pseudorandom number generators (PRNGs). Technically speaking, these only simulate randomness and can exhibit predictable patterns over time if their initial seed is known. Hardware-based RNGs (HRNGs), however, rely on physical processes, such as thermal noise or quantum phenomena, to generate true randomness, making them inherently more secure and unbiased.

While PRNGs are suitable for most applications requiring fairness, critical scenarios like cryptography or scientific simulations often favor HRNGs for their higher standard of unpredictability and fairness. You can also take external inputs for seeds, like the famous lava lamps that encrypt a big part of the internet.

An even better “coin flip” could come from the quantum world. In a quantum state with 50-50% probability, the outcome can only be a truly random one. A good example is the Hong-Ou-Mandel effect, where a photon enters a beam splitter and either gets reflected or transmitted. The relative probabilities are determined by the reflectivity of the beam splitter and it is possible to create a beam splitter so that the photon has equal probability of being reflected or transmitted.

If you are thinking “how on Earth am I going to find a quantum number generator,” well, you are in luck. The Australian National University offers a free quantum number generator that you can use in binary mode or to roll quantum dice or see random colors or images.

Of course, if all that gets too complicated, you can always toss a coin. Just remember not to look at it before, to make it fair.

The post Fair Coins, Flipping Outcomes, and the Subtle Bias of Physics: What Makes a True 50-50? originally appeared on the HLFF SciLogs blog.