[Statistics] Theoretical Probability and Empirical Probability

This post covers types of probability.

1. Introduction

Probability is a branch of mathematics that deals with the study of the likelihood of an event occurring. It is used extensively in many areas of science, engineering, finance, and other fields. Probability is classified into three types: theoretical probability, empirical probability, and subjective probability. Each type of probability is used in different situations and calculated using different methods.

2. Theoretical probability (Classical probability)

Theoretical probability, also known as classical probability, is the probability that is based on a theoretical analysis of the events that could happen in a given situation. The theoretical probability is calculated by dividing the number of ways that an event can occur by the total number of possible outcomes. Theoretical probability is often used in situations where there is no data available to calculate the empirical probability.

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One of the most intuitive examples of theoretical probability is flipping coins. Normally, we assume that the probability of getting head and tail is the same. What is the probability of getting a head for one flip of coin? We know there are two possible outcomes, head and tail, and there is only one favorable outcome, head, so the answer is 1/2.

For theoretical probability, it is assumed that the likelihood of each sample outcome is equally likely. However, the problem is that the probability of each sample outcome is not always equal in reality.

3. Empirical probability (Experimental probability)

Empirical probability, also known as experimental probability, is the probability that is based on actual experiments or observations. The empirical probability is calculated by dividing the number of times an event occurred by the total number of trials. Empirical probability is often used in situations where there is data available to calculate the probability.

Let's use the same example as above, flipping coins. With the empirical probability approach, we try to figure out the probability of getting head by numerous iteration of the experiment (flipping the coin). For example, we might only get tails for the first six experiments. As we flip the coin more and more, we will find that the probability converges to 1/2.

Unlike theoretical probability, empitical probability does not assume that the sample outcomes share the same probability. Therefore, empirical probability is effective when the probability of sample outcomes are not equally likely. Suppose there is a coin which is designed a unique way that it is not reasonable to assume that the probability of getting the head and the tail is the same. In this case, it is inappropriate to use the classical probability since the 'equally likely' assumption is not met.

Nevertheless, it is sometimes diffucult or simply impossible to conduct multiple experiments. Further, it can be unclear how many times of iteration is enough to conclude the probability.

4. Subjective Probability

Subjective probability is the probability that is based on personal judgment or belief. It is often used in situations where there is no data available to calculate the theoretical or empirical probability. Subjective probability is based on personal knowledge, experience, and judgment.

For example, if you are trying to estimate the probability of it raining tomorrow, you might use your own personal judgment and experience to come up with a subjective probability. Subjective probability is used in many areas, including decision-making, forecasting, and risk assessment.

5. Conclusion

In conclusion, theoretical probability is based on theoretical analysis, empirical probability is based on actual experiments or observations, and subjective probability is based on personal judgment or belief. Each type of probability is used in different situations and calculated using different methods. Understanding the different types of probability can help you make better decisions and solve problems in many areas of science, engineering, finance, and other fields.

Empirical probability, also known as experimental probability, is the probability that is based on actual experiments or observations. The empirical probability is calculated by dividing the number of times an event occurred by the total number of trials. Empirical probability is often used in situations where there is data available to calculate the probability.