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 existing everywhere, the use of probability theory allows for 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, where there can be two possible options: 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 integrate uncertainty into their research models as a way of describing 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.
Practical Uses for Probability Theory
Probability is commonly used by data scientists to model situations where experiments, conducted during similar circumstances, yield different results (as in the case of throwing dice or a coin).
Similar approaches have been taken in genetic science, where assessing the likelihood of a genetic disease is tied to frequency of occurrence as opposed 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 a prevailing treatment standard.
An example here is testing the efficacy of a new vaccine, such as the poliomyelitis testing done for the Salk vaccine in 1954 involving almost two million children. Organized by the U.S. Public Health Service, the vaccine nearly eliminated polio as a health problem in the industrialized world.
Also known as the axiomatic method, this type of probability involves a set of axioms (rules) attached to it. 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 singular event in comparison 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.
When using the subjective approach, probability is the likelihood of something happening 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.
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 unlock key factors denoting the relationship between exposures and health risks. The aim here is to quantify uncertainty. This knowledge can fuel a course of action based on 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 the amount of money that needs to be set aside to pay for future losses.
Then there’s the small-business world where owners cannot always turn to their hunches and instincts to run a successful company. In today’s competitive business environment, probability analysis can provide entrepreneurs with key metrics pointing the way to 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 with $500,000 at the low end and $750,000 at 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 nearer to the midpoint of the extremes.
A weather forecast serves as another example of probability theory. The probability for precipitation or severe weather is tied to a specific geographic location. As a result, forecasting can be viewed as the combination of the chance of a weather occurrence and the coverage of that event. 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.”
Advantages and Disadvantages of Probability Theory
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 approach to real-world examples that is easy to digest for those not possessing a math or science background.
With respect to limitations, the classical approach is unable to handle projects where an infinite number of possible outcomes exist. 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 the ability of this approach to handling more complicated tasks.
Unlike the classical approach, relative frequency offers the advantage of being able to handle scenarios where outcomes have different theoretical probability (or likelihood) of occurring. This approach can also manage a probability situation where 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 possessing an infinite number of times cannot be analyzed with 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 level of belief to make 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, and thus changing the classification can change the probability of the event.
For example, if you want to determine the probability of a person contracting an infectious disease like COVID-19, we need to begin with 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.
How Data Scientists Use Probability Theory
Probability allows data scientists to assess the certainty of outcomes of a particular study or experiment. An experiment is a planned study that is executed under controlled conditions. When a result is not already predetermined, the experiment is referred to as 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.
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