Why Improbable Events Happen And Why They Matter

 by Belinda Chau

Imagine a married couple who just found out that they have taken a photograph together at the amusement park when they were kids at a time when they had no idea who each other were. Such coincidence seems so unreal as if it is part of a fictional story where two lovers are destined to be with each other from the beginning. Despite it being a remarkable occurrence worth celebrating, a statistician would argue it is perfectly normal for an improbable event such as this to happen because we should not be surprised to see anything occurring if they have a probability larger than zero.

Like how the way the world works can all be explained by science, improbable events can be explained by probability. Often there is a misconception which incorrectly assumes that improbable events could never be observed. However, there is a line that draws a distinction between improbability and impossibility, and that is if given enough time and opportunity, any seemingly unrealistic but statistically possible outcome could happen. This concept is defined by the law of truly large numbers, one of the five laws that the statistician David Hands set out to provide logical explanations for improbable events and argue why it is more unlikely for improbable events to not occur. The law of inevitability, the law of probability lever, the law of near enough and the law of selection are the other four laws which together form the framework of the improbability principle. In this article, I would explore a few.

According to the law of truly large numbers, if we run a large enough number of independent samples, any possible outcome could take place. For instance, in the 2009 Bulgarian state lottery, two consecutive draws happened to have the exact same set of winning numbers: 4, 15, 23, 24, 35 and 42. The odds of this happening was one in four million as if something suspicious was going on. However, we can think about it in another way: although we do not expect to see it happening, each combination has the exact same probability of winning, so such a coincidence should not be surprising given the number of draws played already was so big - two draws every week for twenty-six years. People have different ways of buying lottery tickets which they believe would maximise the likelihood of them winning such as predicting what would happen by studying trends of the past draws, not buying numbers that are too close to each other or not buying the set of combination that has appeared before, but the most important things forgotten about are that every draw are independent and every number has equal chance of coming up, so really winning all comes down to luck.

Next, we have the law of probability lever which states that a slight change to circumstances could make the improbable event more likely; the reason for this is because a wrong model, which makes the wrong assumption about the reality, is being used, therefore the actual probability of the event is higher than it seems. For instance, the Titanic was said to be unsinkable because waterproof barriers were installed between the twenty compartments, so that the barriers could act as a good line of defence if any compartment got flooded. Then - as we know what happened - the ship crashed into an iceberg which created a large hole on the side of the ship and damaged five neighbouring compartments. It totally changed the probability of sinking because the waterproof barriers were futile in this situation. The ship was not engineered to withstand such a potent attack. When one compartment got damaged, it was very likely that the other neighbouring compartments got damaged too, forming this deadly chain effect. Calculating probability can require very careful thoughts about all the factors interfering with the outcome. In this example, we see the tragic consequence resulting from the wrong model used which did not factor in possible changes to circumstances. Who would have thought the ship could crash into the iceberg? No matter how improbable it seemed, it was still a possibility we needed to be cautious of, so it was crucial to make sure the ship was well-prepared for it.

While a ship could be engineered to minimise the likelihood of an accident, in the economic world, often highly improbable events end up happening and have a significant impact on the economy which we cannot avoid (the impact can be both positive and negative though). They are called black swan events which cannot be predicted with any standard statistical tools such as extrapolation or normal distribution all because of their rarity, so there is barely any data available for us to draw inferences from.

The Covid-19 pandemic could be seen as a black swan event as this virus suddenly put the economy through a severe recession as though everything started going backward. In 2020, the UK experienced a decline in GDP by 9.7% as protective measures were put in place which greatly limited the amount of economic activities that could still be running. If we zoom in on the change in UK GDP during the first lockdown, the figure was 25% lower in April 2020 than in February 2020. Viewing improbability from an economic perspective demonstrates why we cannot get improbability mixed up with impossibility because that would make us vulnerable to shocks and leave us in a state of panic struggling to sort out the situation. Being prepared by strengthening the economy and promoting sustainable economic growth could help us make a strong recovery even if a black swan event hits us.

In conclusion, the occurrence of improbable events is higher than we imagine. There are countless improbable events that could be happening around the world and some can hold destructive power which we cannot underestimate. If something harmful is a possibility, no matter how improbable, we need to see it as significant and come up with solutions to combat it instead of betting on it not happening.