# “The History and Science of Coin Flipping”

Whether you are watching a high-stakes football match or are observing someone trying to make a decision, you may come across one behaviour that is a catalyst for future action. In fact, it is such a normal part of human discourse that we often ignore it. It fades into the background as we focus on other pressing needs.

Specifically, we are talking about the coin flip.

It's such a simple thing. You flip the coin and there is a binary outcome. You get immediate feedback and then you move on with your life.

But having said this, what seems like one of the world’s simplest behaviours overshadows some of the insights contained within a coin flip. Every coin flip comes embedded with centuries of history, including the history of statistics, probability, and the law of averages.

So even though your friends may half-heartedly observe a coin toss and think that it is mundane and boring, we encourage you to think twice. One coin flip can tell you so much about human history. With those insights in mind, you can adopt a new perspective toward the world around you.

The Genesis of Transformative Insights

Coin flipping itself dates back all the way to Ancient Rome. The Romans called coin flipping “navia aut caput,” which roughly translates to “ship or head.” Some Roman coins had ships on one side and the head of an emperor on the other, signalling that the Romans at least thought about flipping their currency. Along with the Romans, citizens in medieval England were tossing their coins in a game called Cross and Pile. Again, the cross and pile represented the heads and tails that we know today.

From that point forward, we have used the coin toss to entertain ourselves, make decisions, and even decide elections. But why is it that the simple act of tossing a coin is seen as an objective, deterministic action that can be the final arbiter in our day-to-day lives?

The answer comes down to probability and statistics. As you likely know, the toss of a coin involves a binary outcome. The coin can be either heads or tails (for the sake of this discussion, we are assuming that the coin is fair. We are also excluding the slim chance that a coin ends up stuck on its side or gets stuck in the ground). Therefore, the probability of a coin landing heads (or tails) is 50%. This objective, binary outcome is why coin flips are so frequently used to break ties or make decisions. So long as the coin toss is fair, each side has an equal chance of winning on an individual coin toss. It is a neutral, independent arbiter that has no opinions or preconceived notions of either party.

Extrapolating this further, we would naturally expect that for every 100 tosses of the coin, 50 would emerge heads and 50 tails. You may think: “every toss has a 50% chance of heads and a 50% chance of tails, so that means I’ll get 50 heads and 50 tails every time, right?”

It doesn’t necessarily work this way in real life. In fact, you may flip 55 heads and 45 tails, or vice versa. Out of 100 tosses, there may be a 70-30 split—or even something more lopsided.

Probability and statistics can explain this phenomenon. Even though every individual coin flip has a 50% chance of heads and a 50% chance of tails, there is a chance—no matter how small—that you will receive a skewed outcome from a defined number of flips. As just one example, let’s say that you are flipping a coin four times. While each coin flip has an equal chance of resulting in heads or tails, the probability of achieving two heads and two tails is not 50-50. Instead, there is a 37.5% chance of that occurring.

Why is this?

With four flips of a coin, there are 16 possible outcomes. Specifically,, there is one way to get zero heads, four ways to get one heads, six ways to get two heads, four ways to get three heads, and one way to get four heads. To obtain the probabilities, we divide the number of ways by the total number of outcomes (16). Therefore, out of four coin flips, there is a 6.25% chance of obtaining zero or four heads, a 25% chance of obtaining one or three heads, and a 37.5% chance of obtaining two heads.

This is a simple illustration of how several coin flips may lead to skewed results in the short run. Even though there is a slim chance (6.25%) of flipping all heads or all tails, this does not equal a 0% chance. Further, things like the Gambler’s Fallacy may lead us astray. They may cause us to believe that if we see more heads or tails in a successive period, we will see a run of tails or heads in the near future in order to “balance out”the tosses. Coin tosses are independent, however, and have no relationship to successive or subsequent losses.

The danger of a small sample size is that it may contain a significant amount of noise. However, as the sample size increases and potential outcomes become exponentially higher, the law of large numbers works in your favour. Essentially, the law of large numbers states that as a sample size grows, the mean of that sample size converges to the average of the whole population. In other words, as you flip more and more coins, you will discover that your proportion of heads and tails results will be approximately equivalent to each other. To put it another way, as we flip more coins, a graphical representation of the flips looks like a bell curve.

This discussion has countless applications in our lives—even if we aren’t flipping coins. While we don’t encounter many situations where there is an even probability of a binary outcome, things like the law of large numbers and Gambler’s Fallacy come up in our lives. As a basic example, let’s say you are on holiday and visit a casino. You are playing roulette and are betting that a red number will appear on every turn of the roulette wheel. While you may be winning in the short-term, the law of large numbers is working against you, as there is a 47.4% chance that any turn will land on a red or black number. If you adopt this roulette strategy for a significant amount of time, you will end up losing money rather than winning money. Moreover, you may be tempted to bet on black after a series of red turns, but the Gambler’s Fallacy is convincing you that turns are dependent, not independent. Even though this is just one example, the fact remains that these insights from coin flips can subtly affect our lives—including our pocketbooks.

Compelling Insights From a Simple Act

The physical act of tossing a coin is extremely simple. You just take the coin, toss it in the air, and get immediate feedback. This simple act, however, has led to a variety of critical insights about probability and statistics. With these insights in hand, we can adopt a more nuanced, appreciative view of our world today.