# Empirical Probability – Definition, Application, and Examples

Empirical probability is an important statistical measure that utilizes historical or previous data. It reflects the measure of how likely a certain outcome can occur given the number of times this particular event has occurred in the past.

Empirical probability is also applied in the real world – making it an important statistical tool when analyzing data in finance, biology, engineering and more.

When calculating empirical probability, count the number of times that the favorable outcome has occurred and divide it by the total number of trials or experiments. This is essential when studying real-world and large-scale data.

This article covers all the fundamentals needed to understand what makes empirical probability unique. We’ll also show you examples and word problems that involve empirical probability. By the end of this discussion, we want you to feel confident when calculating empirical probabilities and solving problems involving them!

## What Is Empirical Probability?

Empirical probability is a number that represents the calculated probability based on the resulting data from actual surveys and experiments. From its name, this probability depends on the empirical data that is already available for assessment.

This is why empirical probability is classified as an experimental probability as well.

\begin{aligned}\textbf{Experimental Probability} &= \dfrac{\textbf{The Number of Times a Certain Event has Occurred}}{\textbf{Total Number of Trials Conducted for the Experiment}} \end{aligned}

From the formula shown above, the empirical probability (represented as $P(E)$) is dependent on two values:

1. The number of times that a specific or favorable outcome has occurred
2. The total number of times that the experiment or the event has occurred

Probabilities can either be empirical or theoretical, so to better understand the concept of empirical probability, let’s observe how these two classifications differ. To highlight their difference, imagine tossing a six-faced die and predicting the probability of getting an odd number.

 Theoretical Probability Empirical Probability A six-faced die will have the following numbers: $\{1, 2, 3, 4,5, 6\}$.This means that there are three odd numbers out of six. The theoretical probability (represented by $P(T)$) would be equal to:\begin{aligned}P(T) &= \dfrac{3}{6}\\&= \dfrac{1}{2} \end{aligned} Suppose that in an experiment where the die was tossed $200$ times, odd numbers appeared $140$ times.Empirical probability depends on past data, so from this, we expect odd numbers to appear with an empirical probability of:\begin{aligned}P(T) &= \dfrac{140}{200}\\&= \dfrac{7}{10} \end{aligned}

This example shows that theoretical probability bases its calculations on the expected number of outcomes and events.

Meanwhile, empirical probability is affected by the result of previous trials.

This is why empirical probability has its disadvantages: the accuracy of probability depends on the sample size and may reflect values far from the theoretical probability. Empirical probability does have a wide list of advantages as well.

Since it is dependent on historical data, it is an important measure when predicting the behavior of real-world data in research, financial markets, engineering and more. What makes empirical probability great is that all hypotheses and assumptions are backed by data.

Seeing the importance of empirical probability and its applications, it’s time that we learn how to calculate empirical probabilities using given data or experiments.

## How To Find Empirical Probability?

To find the empirical probability, count the number of times the desired outcome has occurred then divide this by the total number of times the event or trial has occurred. The empirical probability can be calculated by the formula shown below.

\begin{aligned}\boldsymbol{P(E)} = \boldsymbol{\dfrac{f}{n}}\end{aligned}

For this formula, $P(E)$ represent the empirical probability, $f$ represent the number of times or frequency that the desired outcome occurred, and $n$ represent the total number of trials or events.

 Result After Tossing the Coin Eight Times Experiment Number 1 2 3 4 5 6 7 8 Resulting Face Tail Head Tail Head Head Tail Tail Tail

Suppose that an unbiased coin is tossed eight times and the result is recorded as shown by the table above. Now, to calculate the empirical probability of getting tails, we count the number of times the coin landed on tails.

Divide this number by the total number of trials, which for our case is equal to $8$. Hence, the empirical probability is as shown below.

\begin{aligned}f_{\text{Tails}}&= 5\\n&= 8\\P(E)&= \dfrac{f_{\text{Tails}}}{n}\\&= \dfrac{5}{8}\\&= 0.625\end{aligned}

This means that from the result of tossing the coin eight times, the empirical probability of getting tails is $0.625$. Apply the same process to calculate the empirical probability of the coin landing on heads.

Of course, we know that the theoretical probability of a coin landing on its head and on its tail are both equal to $\dfrac{1}{2} = 0.50$. By adding more trials in the experiment, the empirical probabilities of getting either a head or a tail will approach this value as well.

In the next section, we’ll try out different problems and situations where empirical probability is involved. When you’re ready, jump down and join the fun below!

### Example 1

Suppose that a die is tossed ten times and the table below summarizes the result.

 Result After Tossing the Die Ten Times Experiment Number 1 2 3 4 5 6 7 8 9 10 Resulting Face 6 4 2 1 1 2 3 5 4 5

If we base our empirical probability on this result, what is the experimental probability that when the die is tossed, the die shows $5$?

### Solution

If we base our calculations on the table shown above, let’s count the number of times that the die has shown $5$. Divide this number by $10$ since the die was tossed ten times for this experiment.

\begin{aligned}f_{\text{5}}&=2\\n&= 10\\P(E)&= \dfrac{f_{\text{5}}}{n}\\&=\dfrac{2}{10}\\&= 0.2\end{aligned}

This means that from the experiment, the empirical probability of getting a $5$ is $0.2$.

### Example 2

Monica is conducting a survey determining the number of morning people and night owls in her dormitory. She asked $100$ residents whether they’re more productive in the morning or at night. She found out that $48$ residents are more productive in the morning. What is the empirical probability that Monica meets someone who is a night owl?

### Solution

First, let’s find out the number of residents who identify themselves as night owls. Since Monica asked $100$ residents and $48$ of them are more productive in the morning, there are $100 – 48 = 52$ residents who identify as night owls.

Calculate the empirical probability by dividing the number of reported night owls over the total number of residents that were surveyed by Monica.

\begin{aligned}f_{\text{Night Owl}}&= 52\\n&= 100\\P(E)&= \dfrac{f_{\text{Night Owl}}}{n}\\&=\dfrac{52}{100}\\&= 0.52\end{aligned}

This means that the empirical probability of meeting a night owl in Monica’s dormitory is $0.52$.

### Example 3

Suppose that we use the same table from the previous question. If Monica’s dormitory has a total of $400$ residents, how many residents are more productive in the morning?

### Solution

Using the table from Example 2, calculate the empirical probability of meeting a morning person in the dormitory by dividing $48$ by the total number of residents surveyed by Monica.

\begin{aligned}f_{\text{Morning Person}}&= 48\\n&= 100\\P(E)&= \dfrac{f_{\text{Morning Person}}}{n}\\&=\dfrac{48}{100}\\&=0.48\end{aligned}

Utilize the empirical probability of finding a morning person to approximate the number of residents who are more productive in the morning. Multiply $0.48$ by the total number of residents.

\begin{aligned}f_{\text{Morning Person}} &= P(E) \cdot n\\&= 0.48 \cdot 400\\&= 192\end{aligned}

This means that there are approximately $192$ residents who are more productive in the morning.

### Practice Questions

1. Suppose that a die is tossed ten times and the table below summarizes the result.

 Result After Tossing the Die Ten Times Experiment Number 1 2 3 4 5 6 7 8 9 10 Resulting Face 6 4 2 1 1 2 6 4 4 5

If we base our empirical probability on this result, what is the experimental probability that when the die is tossed, the die shows $4$?

A. $0.17$
B. $0.20$
C. $0.25$
D. $0.30$

2. Using the same table from the previous problem, what is the experimental probability that when the die is tossed, the die shows $3$?

A. $0$
B. $0.20$
C. $0.24$
D. $1$

3. Jessica runs a breakfast buffet and noted that out of $200$ customers, $120$ prefer pancakes over waffles. What is the probability that a customer prefers waffles?

A. $0.12$
B. $0.40$
C. $0.48$
D. $0.60$

4. Using the same data from the previous problem, how many customers are expected to prefer pancakes if Jessica has a total of $500$ customers in a day?

A. $200$
B. $240$
C. $300$
D. $480$

5. There are four books with different genres: Thriller, Nonfiction, Historical Fiction and Sci-Fi. These books are then covered and one book is randomly picked each time for $80$ times. The table below summarizes the result:

 Genre Thriller Historical Fiction Sci-Fi Nonfiction Number of Times Picked 24 32 18 26

What is the empirical probability of randomly picking a book with historical fiction as the genre?

A. $0.32$
B. $0.40$
C. $0.56$
D. $0.80$

6. Using the same result and table from the previous item, if $400$ students are asked to randomly pick a book, how many will have thriller as the book’s genre?

A. $120$
B. $160$
C. $180$
D. $220$