# Empirical Probability: Definition and How to Calculate It

By Indeed Editorial Team

Published May 31, 2022

The Indeed Editorial Team comprises a diverse and talented team of writers, researchers and subject matter experts equipped with Indeed's data and insights to deliver useful tips to help guide your career journey.

Experimental probability can be a useful statistic for calculating the chance of an event happening. This statistical metric may be beneficial in a variety of financial, technological, and commercial scenarios when more precise assumptions about an occurrence are necessary. Learning about empirical probability can help you determine the likelihood of an event occurring.

In this article, we define empirical probability, review its formula, discuss how to calculate it, differentiate between theoretical and experimental probability, explore how this ratio differs from theoretical probability, review other types of probability, and provide an example of its calculation.

## What is empirical probability?

In statistics and scientific research, empirical probability is the analysis and manipulation of data derived from research findings on an outcome seen during experimental trials. This probability represents an estimation of the likelihood of an event occurring based on the frequency with which it occurs in experimental trials. The primary benefit of this kind of probability is that people consider it to be devoid of assumptions. This means they don't assume any data or hypotheses. You can also refer to this kind of probability as relative frequency.

Each observation you make when conducting tests or doing probability calculations is a separate trial. Statisticians, researchers, analysts, and business and finance professionals may calculate the experimental probability of an event occurring to ascertain the potential gains associated with innovation and investments that may involve risk.

Read more: How to Conduct a Risk Assessment (Tips and Definition)

## What is the formula for experimental probability?

To calculate the probability of an event or outcome occurring, you can use the formula:

P(E) = number of times an event occurs / total number of trials.

The P(E) denotes the experimental probability, while the number of times an event occurs denotes the number of times you accomplish a certain result during each trial. The term total number of trials refers to the number of times an experiment undergoes repetition to obtain the desired conclusion. For example, if you want to get the experimental chance of lightning hitting the same site several times, you can first determine the number of times lightning has already struck the location. Then you can divide it by the number of times you observe lightning striking the location.

## Calculating experimental probability

Understanding the connection between a previous event and its likelihood of recurrence in the future may assist you in making critical financial and investment decisions. You may use the experimental probability formula in the following manner:

The experimental probability quantifies the chance of an event happening based on its previous occurrences. Consequently, it's critical to determine the frequency with which you notice the occurrence or consequence occurring throughout your trials. For instance, if a financial analyst wishes to estimate the likelihood of earning a ROI, they may calculate the times an instrument had excellent results for previous investors.

After determining the number of times your desired outcome happens, you can divide it by the number of trials conducted throughout your study. For instance, in the financial analyst's scenario, the number of trials may equal the number of years they anticipate receiving the average return. Here is an example of how a professional can apply the formula if they anticipate receiving the same average return over the following decade based on prior data assessments:

Example: A financial analyst finds that an investment averages a \$250,000 yearly return and wants to calculate the experimental likelihood of similar returns over the following 10 years. If the investment instrument generated \$250,000 annually over the previous seven years, the analyst can decide that the result of the \$250,000 return happened seven times before. Using this information, the analyst calculates the experimental probability:
Number of times the outcome occurs: 7
Total number of trials: 10
Formula: 70% = 100% x 0.7 = 7 / 10 This shows that the empirical chance of the event of a \$250,000 return happening within the analyst's time period is 70%. Depending on the analyst's assessment of the client's unique business objectives, they may suggest the investment opportunity because of its high likelihood of returns.

Read more: Understanding How to Complete a Risk Analysis

## Theoretical vs. experimental probability

Theoretical probability makes assumptions about a larger population's data set. Additionally, the theoretical probability is calculable with no real experiments. Rather than that, you use logical thinking and your knowledge of the circumstance to determine the possibility of an event happening. You can only use theoretical probability to compare your predicted results to the total number of possibilities.

Empirical probabilities focus on experiments and actual observations to determine the likelihood of occurrences. This probability type also uses historical facts rather than assumptions for constructing the numbers that comprise the experimental probability formula. While calculating theoretical probability involves the same division processes as other probability formulas, you divide the number of favourable outcomes by the number of all possible outcomes.

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## Other types of probability

Probability is the discipline of mathematics concerned with the occurrence of a random event, and there are four distinct types, classical, empirical, subjective, and axiomatic. Here's some more information about them:

### Classical

According to the classical or theoretical view of probability, if there are X equally probable possibilities and event Y contains precisely Z of these events, the probability of Y is Z divided by X. This is often the first viewpoint individuals see throughout their education. For instance, when rolling a fair die, there are six equally likely outcomes, so there is a 1/6 probability of rolling each number.

While this approach has the benefit of being conceptually straightforward, most times, it has limitations, because not all situations offer an infinite number of equally plausible alternatives. For example, rolling a weighted dice produces a limited number of results that aren't equally probable, yet monitoring employee earnings over a long period and into the future produces an infinite number of potential outcomes for their greatest prospective earnings.

Read more: 18 Data Analyst Skills for Success

### Subjective

Subjective probability considers an individual's own opinion or judgment that an event might occur. For instance, an investor may have an informed understanding of the market and instinctively discuss the likelihood of a certain company increasing in value tomorrow. You can logically know how that subjective perspective corresponds to theoretical or experimental perspectives. Subjective probability is the possibility that a person's knowledge and emotions may lead them to predict the outcome accurately, regardless of whether they use formal calculations.

While many professions depend on fact-based approaches for probability calculation, subjective probability may be valuable as well. Subjective probability has a wide range of applications in finance and statistics, but it's particularly effective when dealing with unknown variables, estimation, and forecasting.

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### Axiomatic

Axiomatic probability is a unifying approach to probability in which the coherent requirements utilized in theoretical and experimental probability establish subjective probability. You can apply Kolmogorov's set of principles or axioms to many types of probability. Mathematicians refer them to as Kolmogorov's three axioms. When utilizing axiomatic probability, you may quantify the likelihood of an event happening or not occurring. In daily life, this probability theory is useful for risk assessment and modelling. The insurance sector and financial markets use actuarial science to calculate pricing and make decisions.

## Example of calculating experimental probability

Here is an example of calculating this kind of probability:

Tech-Driven Solutions, Inc. is creating a projection for its investment returns over the next five years so company executives can understand how the new investment plan can be beneficial. Financial analysts and planners at Tech-Driven Solutions apply an empirical analysis to determine the following financial statistics, historical data about the investment instrument indicate average returns of \$300,500 annually. The investment instrument produced this average ROI for the past three years. Tech-Driven Solutions is measuring a period of five years.

Using this data, analysts determine that the number of times the return average occurs is for the past three years, and the total number of trials becomes the period analysts forecast. Here, Tech-Driven Solutions wants to know the likelihood of the investment producing similar results over five years, or for five trials. Financial analysts and planners use the experimental probability formula and the empirical analysis data:

Number of times the outcome occurs: 3
Total number of trials: 3
Formula: 60% = 0.6 = 3 / 5

The analysts deduce that the investment opportunity has a 60% likelihood of producing an average annual return of \$300,500 over the next five years.