# Empirical Probability

A type of probability that is based on historical data

## What is Empirical Probability?

Empirical probability, also known as experimental probability, refers to a probability that is based on historical data. In other words, empirical probability illustrates the likelihood of an event occurring based on historical data.

### Formula for Empirical Probability

Where:

• Number of Times Occurred refers to the number of times a favorable event occurred; and
• Total No. of Times Experiment Performed refers to the total amount of times the event was performed.

### Example of Theoretical Probability

#### Example 1

The table below shows a dice thrown three times and the corresponding result. What is the empirical probability of rolling a 4?

Empirical Probability = 0 / 3 = 0%. The empirical probability of rolling a 4 is 0%.

#### Example 2

The table below shows a coin toss three times and the corresponding result. What is the empirical probability of getting a head?

Empirical Probability = 3 / 3 = 100%. The empirical probability of getting a head is
100%.

#### Example 3

In a buffet, 95 out of 100 people chose to order coffee over tea. What is the empirical probability of someone ordering tea?

Empirical Probability = 5 / 100 = 5%. The empirical probability of someone ordering tea is 5%.

### Advantages and Disadvantages

The main advantage of using empirical probability is that the probability is backed by experimental studies and data. It is free from assumed data or hypotheses. However, there are two big disadvantages of empirical probability to consider:

#### 1. Drawing incorrect conclusions

Using empirical probability can cause wrong conclusions to be drawn. For example, we know that the chance of getting a head from a coin toss is ½. However, an individual may toss a coin three times and get heads in all tosses. He may draw an incorrect conclusion that the chances of tossing a head from a coin toss are 100%.

#### 2. Insufficient sample size

Small sample sizes reduce accuracy. Therefore, large sample sizes are generally used for empirical probability to attain a good probability representation. For example, if an individual wanted to know the probability of getting a head in a coin toss but only used one sample, the empirical probability would be either 0% or 100%.

### Different Types of Probabilities

Apart from empirical probability, there are two other main types of probabilities:

#### 1. Classical probability

Classical probability (also called a priori or theoretical probability) refers to probability that is based on formal reasoning. For example, the classical probability of getting a head in a coin toss is ½.

#### 2. Subjective probability

Subjective probability refers to probability that is based on experience or personal judgment. For example, if an analyst believes that “there is an 80% probability that the S&P 500 will hit all-time highs in the next month,” he is using subjective probability.

### Related Readings

CFI is the official provider of the global Financial Modeling & Valuation Analyst (FMVA)™ certification program, designed to help anyone become a world-class financial analyst. To keep learning and advancing your career, the additional CFI resources below will be useful:

• Basic Statistics Concepts in Finance
• Bayes’ Theorem
• Forecasting
• Subjective Probability

### Financial Analyst Certification

Become a certified Financial Modeling and Valuation Analyst (FMVA)® by completing CFI’s online financial modeling classes and training program!
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