Written by Sam Broverman, PhD, ASA

“Probability” is a term that is frequently encountered in various contexts. I’m probably going to say some things in this post about “probability”—no, I definitely will! Here, I just used the word in a typically colloquial way, which is how most people use it. Someone using the word in that manner is probably saying that there is a better than 50-50 chance that the event they are referring to will happen, something like “I’m probably staying home tonight.”

Another part of our day-to-day where we meet probability is in weather forecasts. In weather forecasts, the Probability of Precipitation (POP) is provided, measured as a percentage. Typically, the calculation is the forecaster’s confidence in precipitation multiplied by the percentage of the forecast area that is expected to receive precipitation. If the forecaster is “50% sure that there will be precipitation in 60% of the area covered by the forecast,” the forecast will be that there is a 30% chance of precipitation in the area. The weather forecaster bases the forecast on previous weather behavior that had similar conditions to those of the current weather.

Some people gamble on sports events and are given “odds” describing what they would win for a one-dollar bet. If the odds are 7 to 2 that a particular horse will win a specific race, then a bet of one dollar on that horse winning the race will result in the bettor winning $3.50 if that horse wins. (The 7 to 2 odds in this example do not mean that there is a 2 in 7 (about 28%) probability that the horse will win. Odds are sometimes misinterpreted as representing the probability of the event occurring, but they are determined by the amounts bet for and against the horse winning.)

There are numerous examples in everyday life where the concept of “probability” arises. Some are quantified with a specified numerical probability, like weather forecasts. Some are only loosely connected to the concept of probability, such as the odds in the horse race example. Many are indeterminate, like the sentiment that “I’m probably staying home tonight.”

Since ancient times, ideas related to probability have been studied, and a mathematical discipline has been developed that formalizes concepts and methods to measure chance and the likelihood of certain types of events occurring. This is the mathematical discipline that actuarial students study as part of their education and preparation for the actuarial profession.

Actuaries analyze financial risk, with “risk” being a crucial component of the analysis. These financial risks include life insurance benefits, pension benefits, and property insurance benefits, among others. The study of financial risk requires a very thorough understanding of how risk is measured. Actuaries undergo rigorous training and government oversight to ensure they possess the knowledge and capability to assess risk effectively. One aspect of the training is education in mathematical probability and statistics.

In this blog series, I will not intend to introduce the formal study of probability. Instead, I will address concepts that may be confusing or challenging for students who are just beginning to study Actuarial Science.

Let’s tackle the first concept, it being the mathematical basis for probability: randomness. Mathematically, an event is random if its outcome is unpredictable or if it is impossible to know in advance what the result will be. The outcomes of flipping a coin or tossing a die are two classic examples of random events. Some might suggest that if enough information is known about the force with which the coin is flipped, the height from the surface at which it will land, and so on, then it would be possible to predict how the coin will land, in which case its outcome would not be random. In the real world, we don’t have that information, so we would likely agree that the outcome of the coin flip is random.

It could be argued that if enough information about an event is known, its outcome would not be random. For instance, most calculators have a random number generator that generates a number between 0 and 1. There is an algorithm within the calculator that calculates the number, so if the algorithm is known, the number it calculates would be known in advance and, therefore, not random. On the other hand, the algorithm may appear to generate random numbers in the sense that someone observing a large number of successively generated numbers would not be able to discern any pattern. Furthermore, the algorithm might be constructed to generate numbers that are not “biased” in the sense that the proportion between 0 and 0.5 and the proportion between 0.5 and 1 (and other similar proportions) are essentially the same in the long run. The numbers generated are referred to as “pseudorandom numbers.” This is a topic for another blog.

All this is to say that Mathematics exists in its own universe, separate from the real world in which we live. Some aspects of mathematics provide a way of describing the real world, but mathematics is its own world of concepts and relationships.

A coin might be described as a “fair coin,” meaning that when the coin is tossed, the side that will turn up is “equally likely” to be a head or a tail. We have an intuitive idea of what “equally likely" means: half the time the coin is flipped, it will be (we expect it to be) a head and half the time a tail. The mathematical framework that describes the physical phenomenon of a coin toss is called probability theory. It assigns numerical values to the outcomes, representing the “probability” of each possible result. For a fair coin, this would be a probability of 0.5 for a head and the same for a tail.

Theoretical mathematics is based on a carefully constructed foundation that formalizes logic and inference, as well as how mathematical structures, such as integers and real numbers, are defined and relate to one another. Mathematical probability is part of this overall structure, but the study of it for most practical purposes and applications does not require knowledge of all the details of the underlying structure.

In the next blog in the series, I will try to relate the risk and uncertainty of events in the real world (or their representations) to mathematical models. Some models may be better than others. The model of 0.5 probability for a head or a tail in a randomly tossed coin is generally regarded as a very good one. The models used for weather forecasting might not be so highly regarded.