In this lesson, we’re going to explore empirical probability. We’ll become familiar with the definition, look at some examples, and use the formula for empirical probability to become comfortable calculating these types of probabilities.

## What Is Probability?

If you’ve ever wondered how likely it is that a certain event would occur, then you’ve wondered about probabilities. The **probability** of an event is the likelihood that the event will happen. Probability can be expressed as a fraction, decimal, or percentage.

In this lesson, we will concentrate on probability in fraction or decimal form.In decimal form, the range of possible probabilities is from 0 to 1. If an event has **a probability of 0**, then there is no chance of that event happening. If an event has **a probability of 1**, then the event is guaranteed to happen.

The closer the probability of an event is to 1, the more likely the event is to happen, and similarly, the closer the probability of an event is to 0, the less likely it is to happen. For example, an event having probability 0.78 is more likely to happen than an event having probability 0.

14, because 0.78 is closer to 1 than 0.14.

## What Is Empirical Probability?

Let’s concentrate on a specific type of probability called empirical probability. The **empirical probability** of an event is found through observations and experiments.

It is the likelihood that the event will happen based on the results of data collected. For example, suppose we were to survey a group of 50 chefs. We ask them to choose their favorite type of food, such as the Italian, Mexican, French, American, or Chinese choices.If 14 of the chefs chose Chinese as their favorite food, then based on these results, the empirical probability that a chef in this group would prefer Chinese food over all other types of cuisine would be 0.28. This is an example of empirical probability because we carried out an actual observation by collecting data through our survey.

Our probability is based on the results of this data.

## Formula for Empirical Probability

While we were discussing the food ballot, you may have wondered how the probability of a chef choosing Chinese food as their favorite type of cuisine was found. In empirical probability, we have a formula we use to calculate probabilities: we calculate empirical probability by dividing the number of times an event occurred during our experiment or observation by the total number of trials or observations.Looking back at our food example, we found that 14 of the chefs chose Chinese food as their favorite type of cuisine. This is the number of times that the event of Chinese food being chosen as a favorite occurred. We also know that we asked 50 chefs to choose their favorite type of food.

This is the total number of observations. Therefore, our empirical probability formula shows that the probability of a chef in this group choosing Chinese food as their favorite food is 14 / 50. To convert our formula into decimal form, we simply carry out the division: 14 divided by 50 equals 0.28. The results tell us that the probability of us choosing a chef out of this group whose favorite food is Chinese is 0.

28.

## Examples of Empirical Probability

Let’s look at another example. Assume you have a jar of six marbles: one red, one orange, one yellow, one green, one blue, and one purple.You take a marble out of the jar, record the color of the marble, then put the marble back in the jar 30 times. Here are the results.

Based on these results, let’s answer a few questions.If we were to take a marble out of this jar, what is the empirical probability that we would we would get the red marble?From the results of the experiment, we see that we got a red marble eight times out of the 30 trials performed. Therefore, the empirical probability formula shows that the probability of getting a red marble is 8 / 30 = 0.27.

What is the empirical probability that we would get the purple marble?Similarly, our experiment results show that we got a purple marble five times out of the 30 trials, so the empirical probability of getting a purple marble is 5 / 30 or 0.17.Now, if we were to take a marble out of this jar, is it more likely that we would get a red marble or a purple marble?Based on our results, we see that the probability of getting a red marble is 0.27, and the probability of getting a purple marble is 0.17. Since 0.

27 is closer to one than 0.17 is, it is more likely that we would get a red marble based on these results.Now that we have done these examples, why this is an example of empirical probability? This is an example of empirical probability because our probabilities are based on the data we collected by performing our experiment. This is the precise definition of empirical probability.

## Lesson Summary

**Empirical probability** is probability based on data collected through an experiment or observation.

To calculate empirical probabilities, we use the formula for empirical probability. These probabilities are found by dividing the number of times an event occurred in an experiment by the total number of trials or observations. Empirical probability is used in medicine, business, insurance, accounting, and many other areas. Anywhere that we collect data and draw conclusions based on that data, we use empirical probability.