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Experimental Probability – Explanation & Examples

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What is experimental probability?

Practice questions, experimental probability – explanation & examples.

Experimental probability is the probability determined based on the results from performing the particular experiment.

In this lesson we will go through:

The meaning of experimental probability
How to find experimental probability

The ratio of the number of outcomes favorable to an event to the total number of trials of the experiment.

Experimental Probability can be expressed mathematically as:

$P(\text{E}) = \frac{\text{number of outcomes favorable to an event}}{\text{total number of trials of the experiment}}$

Let’s go back to the die tossing example. If after 12 throws you get one 6, then the experimental probability is $\frac{1}{12}$. You can compare that to the theoretical probability. The theoretical probability of getting a 6 is $\frac{1}{6}$. This means that in 12 throws we would have expected to get 6 twice.

Similarly, if in those 12 tosses you got a 1 five times, the experimental probability is $\frac{5}{12}$.

How do we find experimental probability?

Now that we understand what is meant by experimental probability, let’s go through how it is found.

To find the experimental probability of an event, divide the number of observed outcomes favorable to the event by the total number of trials of the experiment.

Let’s go through some examples.

Example 1: There are 20 students in a class. Each student simultaneously flipped one coin. 12 students got a Head. From this experiment, what was the experimental probability of getting a head?

Number of coins showing Heads: 12

Total number of coins flipped: 20

$P(\text{Head}) = \frac{12}{20} = \frac{3}{5}$

Example 2: The tally chart below shows the number of times a number was shown on the face of a tossed die.

a. What was the probability of a 3 in this experiment?

b. What was the probability of a prime number?

First, sum the numbers in the frequency column to see that the experiment was performed 30 times. Then find the probabilities of the specified events.

a. Number of times 3 showed = 7

Number of tosses = 30

$P(\text{3}) = \frac{7}{30}$

b. Frequency of primes = 6 + 7 + 2 = 15

Number of trials = 30

$P(\text{prime}) = \frac{15}{30} = \frac{1}{2}$

Experimental probability can be used to predict the outcomes of experiments. This is shown in the following examples.

Example 3: The table shows the attendance schedule of an employee for the month of May.

a. What is the probability that the employee is absent?

b. How many times would we expect the employee to be present in June?

a. The employee was absent three times and the number of days in this experiment was 31. Therefore:

$P(\text{Absent}) = \frac{3}{31}$

b. We expect the employee to be absent

$\frac{3}{31} × 30 = 2.9 ≈ 3$ days in June

Example 4: Tommy observed the color of cars owned by people in his small hometown. Of the 500 cars in town, 10 were custom colors, 100 were white, 50 were red, 120 were black, 100 were silver, 60 were blue, and 60 were grey.

a. What is the probability that a car is red?

b. If a new car is bought by someone in town, what color do you think it would be? Explain.

a. Number of red cars = 50

Total number of cars = 500

$P(\text{red car}) = \frac{50}{500} = \frac{1}{10}$

b. Based on the information provided, it is most likely that the new car will be black. This is because it has the highest frequency and the highest experimental probability.

Now it is time for you to try these examples.

The table below shows the colors of jeans in a clothing store and their respective frequencies. Use the table to answer the questions that follow.

What is the probability of selecting a brown jeans?
What is the probability of selecting a blue or a white jeans?

On a given day, a fast food restaurant notices that it sold 110 beef burgers, 60 chicken sandwiches, and 30 turkey sandwiches. From this observation, what is the experimental probability that a customer buys a beef burger?

Over a span of 20 seasons, a talent competition notices the following. Singers won 12 seasons, dancers won 2 seasons, comedians won 3 seasons, a poet won 1 season, and daring acts won the other 2 seasons.

a. What is the experimental probability of a comedian winning a season?

b. From the next 10 seasons, how many winners do you expect to be dancers?

Try this at home! Flip a coin 10 times. Record the number of tails you get. What is your P(tail)?

Number of brown jeans = 25

Total Number of jeans = 125

$P(\text{brown}) = \frac{25}{125} = \frac{1}{5}$

Number of jeans that are blue or white = 75 + 20 = 95

$P(\text{blue or white}) = \frac{95}{125} = \frac{19}{25}$

Number of beef burgers = 110

Number of burgers (or sandwiches) sold = 200

$P(\text{beef burger}) = \frac{110}{200} = \frac{11}{20}$

a. Number of comedian winners = 3

Number of seasons = 20

$P(\text{comedian}) = \frac{3}{20}$

b. First find the experimental probability that the winner is a dancer.

Number of winners that are dancers = 2

$P(\text{dancer}) = \frac{2}{20} = \frac{1}{10}$

Therefore we expect

$\frac{1}{10} × 10 = 1$ winner to be a dancer in the next 10 seasons.

To find your P(tail) in 10 trials, complete the following with the number of tails you got.

$P(\text{tail}) = \frac{\text{number of tails}}{10}$

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Experimental Probability

Experimental probability , also known as empirical probability , is a concept in mathematics that deals with estimating the likelihood of an event occurring based on actual experimental results. Unlike theoretical probability , which predicts outcomes based on known possibilities, experimental probability is derived from real-life experiments and observations.

To understand this better, imagine flipping a coin. The theoretical probability of landing heads is 50% or 1/2. However, if you actually flip the coin 100 times and record the outcomes, you might get heads 48 times. The experimental probability of getting heads would then be 48/100 or 0.48.

In this article, we will explore the concept of experimental probability, its significance, and how it differs from theoretical probability. We will discuss the formula for calculating experimental probability, provide examples to illustrate its application.

Table of Content

What is Probability?

What is experimental probability, formula for experimental probability, examples of experimental probability, what is theoretical probability, experimental probability vs theoretical probability.

Solved Examples
Practice Problems

The branch of mathematics that tells us about the likelihood of the occurrence of any event is the probability . Probability tells us about the chances of happening an event.

The probability of any element that is sure to occur is One(1) whereas the probability of any impossible event is Zero(0). The probability of all the elements ranges between 0 to 1.

There are two ways of studying probability that are

Theoretical Probability

Now let’s learn about both in detail.

Experimental probability is a type of probability that is calculated by conducting an actual experiment or by performing a series of trials to observe the occurrence of an event. It is also known as empirical probability.

To calculate experimental probability, you need to conduct an experiment by repeating the event multiple times and observing the outcomes. Then, you can find the probability of the event occurring by dividing the number of times the event occurred by the total number of trials.

The experimental Probability for Event A can be calculated as follows:

P(E) = (Number of times an event occur in an experiment) / (Total number of Trials)

Now, as we learn the formula, let’s put this formula in our coin-tossing case. If we tossed a coin 10 times and recorded a head 4 times and a tail 6 times then the Probability of Occurrence of Head on tossing a coin:

P(H) = 4/10

Similarly, the Probability of Occurrence of Tails on tossing a coin:

P(T) = 6/10

Theoretical Probability deals with assumptions in order to avoid unfeasible or expensive repetition experiments. The theoretical Probability for an Event A can be calculated as follows:

P(A) = Number of outcomes favorable to Event A / Number of all possible outcomes

Now, as we learn the formula, let’s put this formula in our coin-tossing case. In tossing a coin, there are two outcomes: Head or Tail.

Hence, The Probability of occurrence of Head on tossing a coin is

Similarly, The Probability of the occurrence of a Tail on tossing a coin is

Experimental Probability vs. Theoretical Probability

There are some key differences between Experimental and Theoretical Probability , some of which are as follows:

Probability in Maths
Probability Distribution
Bayes’ Theorem

Solved Examples of Experimental Probability

Example 1. Let’s take an example of tossing a coin, tossing it 40 times , and recording the observations. By using the formula, we can find the experimental probability for heads and tails as shown in the below table.

Number of Trail Outcome Number of Trail Outcome Number of Trail Outcome Number of Trail Outcome First H Eleventh T Twenty-first T Thirty-first T Second T Twelfth T Twenty-second H Thirty-second H Third T Thirteenth H Twenty-third T Thirty-third T Fourth H Fourteenth H Twenty-fourth H Thirty-fourth H Fifth H Fifteenth H Twenty-fifth T Thirty-fifth T Sixth H Sixteenth H Twenty-sixth H Thirty-sixth T Seventh T Seventeenth T Twenty-seventh T Thirty-seventh T Eighth H Eighteenth T Twenty-eighth T Thirty-eighth H Ninth T Nineteenth T Twenty-ninth T Thirty-ninth T Tenth H Twentieth T Thirtieth H Fortieth T The formula for experimental probability: P(H) = Number of Heads ÷ Total Number of Trials = 16 ÷ 40 = 0.4 Similarly, P(H) = Number of Tails ÷ Total Number of Trials = 24 ÷ 40 = 0.6 P(H) + P(T) = 0.6 + 0.4 = 1 Note: Repeat this experiment for ‘n’ times and then you will find that the number of times increases, the fraction of experimental probability comes closer to 0.5. Thus if we add P(H) and P(T), we will get 0.6 + 0.4 = 1 which means P(H) and P(T) is the only possible outcomes.

Example 2. A manufacturer makes 50,000 cell phones every month. After inspecting 1000 phones, the manufacturer found that 30 phones are defective. What is the probability that you will buy a phone that is defective? Predict how many phones will be defective next month.

Experimental Probability = 30/1000 = 0.03 0.03 = (3/100) × 100 = 3% The probability that you will buy a defective phone is 3% ⇒ Number of defective phones next month = 3% × 50000 ⇒ Number of defective phones next month = 0.03 × 50000 ⇒ Number of defective phones next month = 1500

Example 3. There are about 320 million people living in the USA. Pretend that a survey of 1 million people revealed that 300,000 people think that all cars should be electric. What is the probability that someone chosen randomly does not like the electric car? How many people like electric cars?

Now the number of people who do not like electric cars is 1000000 – 300000 = 700000 Experimental Probability = 700000/1000000 = 0.7 And, 0.7 = (7/10) × 100 = 70% The probability that someone chose randomly does not like the electric car is 70% The probability that someone like electric cars is 300000/1000000 = 0.3 Let x be the number of people who love electric cars ⇒ x = 0.3 × 320 million ⇒ x = 96 million The number of people who love electric cars is 96 million.

Practice Problems on Experimental Probability

Problem 1: A coin is flipped 200 times, and it lands on heads 120 times. What is the experimental probability of getting heads?

Problem 2: A die is rolled 50 times, and the number 3 appears 8 times. What is the experimental probability of rolling a 3?

Problem 3: In a class survey, 150 students were asked if they prefer reading books or watching movies. 90 students said they prefer watching movies. What is the experimental probability that a randomly chosen student prefers watching movies?

Problem 4: A bag contains 5 red, 7 blue, and 8 green marbles. If 40 marbles are drawn at random with replacement, and 12 of them are red, what is the experimental probability of drawing a red marble?

Problem 5: A basketball player made 45 successful free throws out of 60 attempts. What is the experimental probability that the player will make a free throw?

Problem 6: During a game, a spinner is spun 80 times, landing on a specific section 20 times. What is the experimental probability of the spinner landing on that section?

FAQs on Experimental Probability

Define experimental probability..

Probability of an event based on an actual trail in physical world is called experimental probability.

How is Experimental Probability calculated?

Experimental Probability is calculated using the following formula: P(E) = (Number of trials taken in which event A happened) / Total number of trials

Can Experimental Probability be used to predict future outcomes?

No, experimental probability can’t be used to predict future outcomes as it only achives the theorectical value when the trails becomes infinity.

How is Experimental Probability different from Theoretical Probability?

Theoretical probability is the probability of an event based on mathematical calculations and assumptions, whereas experimental probability is based on actual experiments or trials.

What are some Limitations of Experimental Probability?

There are some limitation of experimental probability, which are as follows: Experimental probability can be influenced by various factors, such as the sample size, the selection process, and the conditions of the experiment. The number of trials conducted may not be sufficient to establish a reliable pattern, and the results may be subject to random variation. Experimental probability is also limited to the specific conditions of the experiment and may not be applicable in other contexts.

Can Experimental Probability of an event be a negative number if not why?

As experimental probability is given by: P(E) = Number of trials taken in which event A happened/Total number of trials Thus, it can’t be negative as both number are count of something and counting numbers are 1, 2, 3, 4, …. and they are never negative.

What are Types of Probability?

There are two forms of calculating the probability of an event that are, Theoretical Probability Experimental Probability

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Chapter 6: Triangles

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Chapter 7: Coordinate Geometry

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Chapter 9: Some Applications of Trigonometry

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Chapter 10: Circles

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Chapter 11: Constructions

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Chapter 12: Areas related to circles

Area of a Circle: Formula, Derivation, Examples Area of a Circle is the measure of the two-dimensional space occupied by a circle. It is mostly calculated by the size of the circle's radius which is the distance from the center of the circle to any point on its edge. The area of a circle is proportional to the radius of the circle. Area of Circle 10 min read
Circumference of Circle - Definition, Perimeter Formula, and Examples The circumference of a circle is the distance around its boundary, much like the perimeter of any other shape. It is a key concept in geometry, particularly when dealing with circles in real-world applications such as measuring the distance traveled by wheels or calculating the boundary of round obj 8 min read
Sector of a Circle Sector of a Circle is one of the components of a circle like a segment which students learn in their academic years as it is one of the important geometric shapes. The sector of a circle is a section of a circle formed by the arc and its two radii and it is produced when a section of the circle's ci 12 min read
Arc Length Formula Arc length is the distance along the curved path of a circle or any part of its circumference. We define arc length as measuring the length of a slice of pizza crust. Arc length is calculated using the simple formula: Arc Length= r × θ where 'r' is the radius of the circle and 'θ' is the angle in ra 8 min read

Chapter 13: Surface Areas and Volume

Surface Area of Cuboid The surface area of a cuboid is the total space occupied by all its surfaces/sides. In geometry, a three-dimensional shape having six rectangular faces is called a cuboid. A cuboid is also known as a regular hexahedron and has six rectangular faces, eight vertices, and twelve edges with congruent, o 12 min read
Volume of Cuboid | Formula and Examples Volume of a cuboid is calculated using the formula V = L × B × H, where V represents the volume in cubic units, L stands for length, B for breadth, and H for height. Here, the breadth and width of a cuboid are the same things. The volume signifies the amount of space occupied by the cuboid in three 8 min read
Surface Area of Cube | Curved & Total Surface Area Surface area of a cube is defined as the total area covered by all the faces of a cube. In geometry, the cube is a fascinating three-dimensional object that we encounter daily, from dice to ice cubes. But have you ever wondered about the total area that covers a cube? This is what we call the surfac 15 min read
Volume of a Cube Volume of a Cube is defined as the total number of cubic units occupied by the cube completely. A cube is a three-dimensional solid figure, having 6 square faces. Volume is nothing but the total space occupied by an object. An object with a larger volume would occupy more space. The volume of the cu 9 min read
Surface Area of Cylinder | Curved and Total Surface Area of Cylinder Surface Area of a Cylinder is the amount of space covered by the flat surface of the cylinder's bases and the curved surface of the cylinder. The total surface area of the cylinder includes the area of the cylinder's two circular bases as well as the area of the curving surface. The volume of a cyli 10 min read
Volume of a Cylinder| Formula, Definition and Examples Volume of a cylinder is a fundamental concept in geometry and plays a crucial role in various real-life applications. It is a measure which signifies the amount of material the cylinder can carry. It is also defined as the space occupied by the Cylinder. The formula for the volume of a cylinder is π 11 min read
Surface Area of Cone Surface Area of a Cone is the total area encompassing the circular base and the curved surface of the cone. A cone has two types of surface areas. If the radius of the base is 'r' and the slant height is 'l', we use two formulas: Total Surface Area (TSA) of the cone = πr(r + l)Curved Surface Area (C 8 min read
Volume of Cone- Formula, Derivation and Examples Volume of a cone can be defined as the space occupied by the cone. As we know, a cone is a three-dimensional geometric shape with a circular base and a single apex (vertex). Let's learn about Volume of Cone in detail, including its Formula, Examples, and the Frustum of Cone. Volume of ConeA cone's v 10 min read
Surface Area of Sphere | Formula, Derivation and Solved Examples A sphere is a three-dimensional object with all points on its surface equidistant from its center, giving it a perfectly round shape. The surface area of a sphere is the total area that covers its outer surface. To calculate the surface area of a sphere with radius r, we use the formula: Surface Are 8 min read
Volume of a Sphere The volume of a sphere helps us understand how much space a perfectly round object occupies, from tiny balls to large planets. Using the simple volume of sphere formula, you can easily calculate the space inside any sphere. Whether you're curious about the volume of a solid sphere in math or science 8 min read
Surface Area of a Hemisphere A hemisphere is a 3D shape that is half of a sphere's volume and surface area. The surface area of a hemisphere comprises both the curved region and the base area combined. Hemisphere's Total Surface Area (TSA) = Curved Surface Area + Base Area = 3πr² square units.Curved Surface Area (CSA) = 2πr² sq 13 min read
Volume of Hemisphere Volume of a shape is defined as how much capacity a shape has or we can say how much material was required to form that shape. A hemisphere, derived from the Greek words "hemi" (meaning half) and "sphere," is simply half of a sphere. If you imagine slicing a perfectly round sphere into two equal hal 6 min read
Volume of Combination of Solids When two or more two solids are combined and the combination comes out useful, a shape that can be found in reality is called a combination of solids. When Solids are taught, the major focus is always on the point of their real-life use and applications, For example, a cylinder can be seen in Pipes 9 min read
Frustum of Cone Frustum of a cone is a special shape that is formed when we cut the cone with a plane parallel to its base. The cone is a three-dimensional shape having a circular base and a vertex. So the frustum of a cone is a solid volume that is formed by removing a part of the cone with a plane parallel to cir 10 min read
Conversion of solids - Surface Areas and Volumes Conversions or changes are now a normal feature of our everyday lives. A goldsmith, for instance, melts a strip of gold to turn it into a gem. Likewise, a kid plays with clay forms it into various toys, a carpenter uses the wooden logs to shape various items/furniture for housekeeping. Likewise, for 4 min read
Surface Areas and Volumes Surface Area and Volume are two fundamental properties of a three-dimensional (3D) shape that help us understand and measure the space they occupy and their outer surfaces. Knowing how to determine surface area and volumes can be incredibly practical and handy in cases where you want to calculate th 10 min read

Chapter 14: Statistics

How to find Mean of grouped data by direct method? Statistics involves gathering, organizing, analyzing, interpreting, and presenting data to form opinions and make decisions. Applications range from educators computing average student scores and government officials conducting censuses to demographic analysis. Understanding and utilizing statistica 9 min read
Shortcut Method for Arithmetic Mean Statistics, in layman's words, is the process of gathering, classifying, examining, interpreting, and finally presenting information in an understandable manner so that one can form an opinion and, if necessary, take action. Examples: A teacher collects students' grades, organizes them in ascending 6 min read
How to Calculate Mean using Step Deviation Method? Step Deviation Method is a simplified way to calculate the mean of a grouped frequency distribution, especially when the class intervals are uniform. In simple words, statistics implies the process of gathering, sorting, examining, interpreting and then understandably presenting the data to enable o 7 min read
Graphical determination of Median A measure of central tendency that determines the centrally located value of a given series is known as the Median. The number of values of the series below and above the given series is always equal. To determine the median value of a given series, it is first managed in increasing or decreasing or 5 min read
Ogive (Cumulative Frequency Curve) and its Types A method of presenting data in the form of graphs that provides a quick and easier way to understand the trends of the given set of data is known as Graphic Presentation. The two types of graphs through which a given set of data can be presented are Frequency Distribution Graphs and Time Series Grap 6 min read

Chapter 15: Probability

Types of Events in Probability Whenever an experiment is performed whose outcomes cannot be predicted with certainty, it is called a random experiment. In such cases, we can only measure which of the events is more likely or less likely to happen. This likelihood of events is measured in terms of probability and events refer to t 13 min read
Dependent and Independent Events Dependent and Independent Events are the types of events that occur in probability. Suppose we have two events say Event A and Event B then if Event A and Event B are dependent events then the occurrence of one event is dependent on the occurrence of other events if they are independent events then 8 min read
Experimental Probability Experimental probability, also known as empirical probability, is a concept in mathematics that deals with estimating the likelihood of an event occurring based on actual experimental results. Unlike theoretical probability, which predicts outcomes based on known possibilities, experimental probabil 8 min read
Probability Theory Probability theory is an advanced branch of mathematics that deals with measuring the likelihood of events occurring. It provides tools to analyze situations involving uncertainty and helps in determining how likely certain outcomes are. This theory uses the concepts of random variables, sample spac 10 min read
CBSE Class 10th Maths Formulas: Chapter Wise Formula and Points Mathematics is one of the most scoring subject in CBSE Class 10th board exam. So Students are advised to prepare well for Math in order to score good marks in CBSE Class 10 board exam. GeeksforGeeks has curated the chapter wise Math formulae for CBSE Class 10th exam. These Formulae include chapters 15+ min read
NCERT Solutions for Class 10 Maths 2024-25: Chapter Wise PDF Download NCERT Solutions for Class 10 Maths are tailored by subject matter experts to assist 10 students in securing top marks in their 10 board exams. As we know mathematics is the highest-scoring subject in the CBSE Class 10th board exam This complete resource includes all questions and answers from the CB 15+ min read
RD Sharma Class 10 Solutions RD Sharma Class 10 Solutions offer excellent reference material for students, enabling them to develop a firm understanding of the concepts covered. in each chapter of the textbook. As Class 10 mathematics is categorized into various crucial topics such as Algebra, Geometry, and Trigonometry, which 9 min read
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