Probability is a quantitative measure of the likelihood of a certain event, expressed as either a percentage or a number between 0 and 1. An event which cannot happen has a probability of 0 (or 0%), while an event which is certain to occur has a probability of 1 (or 100%). Examples of probability are everywhere:
- There is a 30% chance of rain on Wednesday.
- The odds of rolling a 6 on a fair die is 1/6 = 1.6666.
- In the general population, 17 in every 100 men are diagnosed with prostate cancer.
Most of us understand what a simple probability is, but it can be more challenging to capture the intricacies of how probability works. Many of the themes I’ll discuss relate to individuals’ misjudgments of probabilities. Before I give a couple of examples, it’s important to think about two primary types of probabilities.
The objective probability of an event can be thought of as the true probability of an occurrence of an event. Consider flipping a fair coin (“fair” meaning there is an equal chance of the coin landing on heads or tails). If the coin is fair, and cannot land on its side, then there is a 0.5 probability the coin will land on heads, and a 0.5 probability the coin will land on tails. When the objective probability of an event can be determined, it is the probability which should be used. For example, if gambling with a fair die, you would likely want to consider betting with the belief that each number 1 through 6 has a 1/6 probability of coming up (instead of some other probability).
Unfortunately, for many events, the objective probability cannot be easily ascertained. Consider a weather forecast. In the universe on any given day, there is some chance of rain. This probability may vary based on location, time of year, and many other scientific factors (some more easily measurable than others). But there is some objective probability of rain. Now, if any particular meteorologist actually knew the objective probability in his town every day, he’d make quite a name for himself! Alas, since the objective probability cannot be fully known, the meteorologist provides a subjective probability of the event of rain: his best estimation as to the likelihood of rain.
In the absence of knowledge of the objective probability of an event, a subjective probability may be the best we can do. But it is crucial to understand when the probability you are facing is objective or subjective. Mistaking his prediction of a 10% chance of rain for an objective probability can have dire consequences if the true probability is much higher and his subjective probability underestimates the potential for a storm!
When we have sufficient information, mathematically computing an objective probability can be straightforward – but we often make mistakes in thinking about these computations. Consider a common coincidence: we are often amazed when, in a room full of people, we encounter someone who shares our birthday. Clearly that must be some spectacular coincidence capturing a low probability event, right?
Take a more concrete example. In a room of 23 people, suppose each one has an equal chance of having a birthday on any day of the year. What are the chances that some pair share the same birthday? 365 days in the year, 23 people: surely the odds must be low! As it turns out, the probability of at least one pair sharing a birthday is just over 50%! This problem, known as the Birthday Paradox, illustrates the often counterintuitive nature of probability, as well as the difficulty we can have in calculating the probability of an event. Many of us would have guessed that the chances in the above example are much lower than 50% – our subjective probability would be an underestimation.
Random events play a very big role in our daily lives, and probability theory provides the central tools to model the likelihood of these events. In fact, Nassim Taleb has an entire book dedicated to the often hidden role of randomness. Understanding how probability works is essential to making good decisions in an uncertain world, and I hope to dedicate more themes to the ins and outs of probability.
*For a straightforward primer on some probability basics, including mathematics no more complicated than algebra, check out Amir Aczel’s Chance.