Probability theory is a branch of mathematics focusing on the analysis of random phenomena. It is an important skill for data scientists using data affected by chance. 

With randomness present everywhere, probability theory enables the analysis of chance events. The aim is to determine the likelihood of an event occurring, often using a numerical scale of between 0 and 1, with the number “0” indicating impossibility and “1” indicating certainty. 

A classic example of this is a coin toss, which has two possible outcomes: heads or tails. Here, the possibility of flipping a head or a tail on a single toss is 50%. When conducting your own experiment, you may find that the outcomes can vary. But if you continue flipping the coin, the outcome grows closer to 50/50.

Probability plays a vital role in many areas of scientific research. Researchers can incorporate uncertainty into their research models to describe their findings. This allows for a predictive distribution of findings tied to what may have been observed in the past.

Randomness and uncertainty are popular themes tied to probability. In Nassim Taleb’s bestselling books The Black Swan and Fooled By Randomness, the claim is made that rare events typically hold more importance than common ones because their effect size is not as restricted. Also, because of their rarity, results are unlikely to be determined. 

Taleb popularized what he calls a “black swan” event, one that is rare, has a catastrophic impact when it does occur and can be explained in hindsight in a way that leads many to believe that it was actually predictable. 

Probability is commonly used by data scientists to model situations in which experiments, conducted under similar circumstances, yield different results (as in the case of throwing dice or a coin).

It also has many practical uses in the business world. Take, for example, the insurance industry, where actuarial records chart the life expectancy of individuals of a certain age. Instead of predicting what will happen to any one individual, the aim is to capture a collective result encompassing a large number of people. 

Similar approaches have been taken in genetic science, where assessing the likelihood of a genetic disease is tied to its frequency of occurrence rather than to predictions about a specific individual. 

Another common application of probability is also commonly applied in clinical trials where new disease treatments, drugs or surgical treatments are being sought. In assessing whether a treatment can be deemed a success or failure, the clinical trial aims to determine whether the new treatment is more successful than the prevailing treatment standard.

A modern example is monitoring real-world vaccine effectiveness using large health-record datasets. For instance, CDC’s interim analysis of the 2024–2025 COVID-19 vaccines used a test-negative design across 137,543 emergency department/urgent care encounters (VISION) and 26,219 hospitalizations among adults ≥65 (VISION) to estimate the extent to which vaccination reduced COVID-19–associated visits and hospitalizations. 

There are three types of probability commonly used in statistical inference. These are: 

Also known as the axiomatic method, this type of probability is based on a set of axioms (rules). For example, you could have a rule that the probability must be greater than 0.5% in order for it to be valid. 

This involves looking at the occurrence ratio of a single event relative to the total number of outcomes. This type of probability is often used after data from an experiment has been gathered to compare a subset of data to the total amount of collected data. 

In the subjective approach, probability is the likelihood that something will happen based on one’s experiences or personal judgment. Here, there are no formal calculations for subjective probability, for it is based on one’s beliefs, judgment, and personal reasoning.

By way of example, during a sporting event, fans of one team share who they are rooting for. This is based on facts or opinions they personally hold regarding the game, the two teams playing and the odds of the team winning.

Probability theory is a tool employed by researchers, businesses, investment analysts and countless others for risk management and scenario analysis.

Take epidemiology, which is the science of disease distribution. Researchers in this field study disease frequency, assessing how the probability differs across groups of people. A present-day example of this is the use of probability by epidemiologists to assess the cause-effect relationship between exposure and illness to the coronavirus.

Probability theory is often used to identify key factors that characterize the relationship between exposures and health risks. The aim here is to quantify uncertainty. This knowledge can inform a course of action aimed at achieving the best outcomes for those affected by various diseases. 

The actuaries who are often employed in the insurance industry make primary use of probability, statistics and other data science tools to calculate the probability of uncertain future events occurring over a period of time. They then apply other data concepts to determine how much money needs to be set aside to cover future losses. 

Then there’s the small-business world, where owners cannot always rely on their hunches and instincts to run a successful company. In today’s competitive business environment, probability analysis can provide entrepreneurs with key metrics to guide them toward the most profitable and productive paths. This analysis offers a controlled way to anticipate potential results. 

For example, if a business enterprise expects to receive between $500,000 and $750,000 in revenue each month, the graph will begin at $500,000 on the low end and $750,000 on the high end. For a typical probability distribution, the graph will resemble a bell curve, where the least likely outcomes fall nearer the extreme ends of the range and the most likely fall nearer to the midpoint of the extremes.

A weather forecast serves as another example of probability theory. The probability of precipitation or severe weather varies by geographic location. As a result, forecasting can be viewed as the combination of the probability of a weather event and its coverage. According to an information statement of the American Meteorological Society:

“A probability forecast includes a numerical expression of uncertainty about the quantity or event being forecast.  Ideally, all elements (temperature, wind, precipitation, etc.) of a weather forecast would include information that accurately quantifies the inherent uncertainty. Surveys have consistently indicated that users desire information about uncertainty or confidence of weather forecasts. The widespread dissemination and effective communication of forecast uncertainty information is likely to yield substantial economic and social benefits, because users can make decisions that explicitly account for this uncertainty.” 

For data scientists, there are a number of advantages and disadvantages with probability that need to be considered.  

The classical method of probability is used when all probable outcomes have an equal likelihood of happening and every outcome is known in advance. The coin toss example above uses the classical approach to probability. The classical approach offers a simple, easy-to-digest approach to real-world examples for those without a math or science background. 

With respect to limitations, the classical approach cannot handle projects with an infinite number of possible outcomes. It’s also ineffective in scenarios where each outcome is not equally likely, as in the case of tossing a weighted die. These limitations affect this approach’s ability to handle more complex tasks. 

Unlike the classical approach, relative frequency offers the advantage of handling scenarios in which outcomes have different theoretical probabilities (or likelihoods) of occurring. This approach can also handle a probabilistic situation in which the possible outcomes are unknown. 

Although you can use relative frequency probability in more diverse situations and settings than classical probability, it has several limitations. The first limitation to relative frequency involves the problem of “infinite repetitions.” This is where experiments with an infinite number of repetitions cannot be analyzed using this theory. So while a large number of trials can be conducted, that number can’t be infinite. 

Problems that benefit from subjective probability are those that require some degree of belief to be made possible. For example, a candidate who may be down in the polls may use subjective probability to make a case for staying in the race. 

Subjective probability also benefits from what is known as the reference class problem. In a reference class problem, assigning a probability to a certain event might require that event to be classified. That classification can be subjective, so changing it can change the event’s probability. 

For example, if you want to determine the probability of a person contracting an infectious disease like COVID-19, you need to begin by assessing which classes of people are relevant to the problem. It’s here where various reference classes can be established. A broad class such as “all U.S. residents” could be used. Or it could be narrowed down to, say, “all residents of the states of X, Y, and Z, where 80% of the deaths are occurring.” In other words, depending on the reference class chosen, different probabilities will emerge.

Probability allows data scientists to assess the certainty of outcomes of a particular study or experiment. An experiment is a planned study conducted under controlled conditions. When the result is not predetermined, the experiment is called a chance experiment. Conducting a coin toss twice is an example of a chance experiment. 

Today’s data scientists need to have an understanding of the foundational concepts of probability theory including key concepts involving probability distribution, statistical significance, hypothesis testing and regression. Learn more statistics concepts that data scientists use regularly; probability distribution is only one of them.

Job opportunities for computer scientists are projected to increase 20% from 2022 to 2033, according to the Bureau of Labor Statistics (BLS). As organizations seek new solutions to capture and analyze large volumes of data, data scientists will be in high demand across a wide range of sectors and industries worldwide, with a median salary of $140,910 per year. Many of these job opportunities will require a master’s degree in computer science or a related field. Check out online master’s in data science programs and find the best degree for your career goals.

Information last updated: December 2025