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Theoretical vs. Experimental Probability: How do they differ?

Probability is the study of chances and is an important topic in mathematics. There are two types of probability: theoretical and experimental.

So, how to define theoretical and experimental probability? Theoretical probability is calculated using mathematical formulas, while experimental probability is based on results from experiments or surveys. In order words, theoretical probability represents how likely an event is to happen. On the other hand, experimental probability illustrates how frequently an event occurs in an experiment.

Read on to find out the differences between theoretical and experimental probability. If you wonder How to Understand Statistics Easily , I wrote a whole article where I share 9 helpful tips to help you Ace statistics.

Table of Contents

What Is Theoretical Probability?

Theoretical probability is calculated using mathematical formulas. In other words, a theoretical probability is a probability that is determined based on reasoning. It does not require any experiments to be conducted. Theoretical probability can be used to calculate the likelihood of an event occurring before it happens.

Keep in mind that theoretical probability doesn’t involve any experiments or surveys; instead, it relies on known information to calculate the chances of something happening.

For example, if you wanted to calculate the probability of flipping a coin and getting tails, you would use the formula for theoretical probability. You know that there are two possible outcomes—heads or tails—and that each outcome is equally likely, so you would calculate the probability as follows: 1/2, or 50%.

How Do You Calculate Theoretical Probability?

• First, start by counting the number of possible outcomes of the event.
• Second, count the number of desirable (favorable) outcomes of the event.
• Third, divide the number of desirable (favorable) outcomes by the number of possible outcomes.
• Finally, express this probability as a decimal or percentage.

The theoretical probability formula is defined as follows: Theoretical Probability = Number of favorable (desirable) outcomes divided by the Number of possible outcomes.

How Is Theoretical Probability Used in Real Life?

Probability plays a vital role in the day to day life. Here is how theoretical probability is used in real life: 

• Sports and gaming strategies
• Analyzing political strategies.
• Buying or selling insurance
• Determining blood groups 
• Online shopping
• Weather forecast
• Online games

What Is Experimental Probability?

Experimental probability, on the other hand, is based on results from experiments or surveys. It is the ratio of the number of successful trials divided by the total number of trials conducted. Experimental probability can be used to calculate the likelihood of an event occurring after it happens.

For example, if you flipped a coin 20 times and got heads eight times, the experimental probability of obtaining heads would be 8/20, which is the same as 2/5, 0.4, or 40%.

How Do You Calculate Experimental Probability?

The formula for the experimental probability is as follows:  Probability of an Event P(E) = Number of times an event happens divided by the Total Number of trials .

If you are interested in learning how to calculate experimental probability, I encourage you to watch the video below.

How Is Experimental Probability Used in Real Life?

Knowing experimental probability in real life provides powerful insights into probability’s nature. Here are a few examples of how experimental probability is used in real life:

• Rolling dice
• Selecting playing cards from a deck
• Drawing marbles from a hat
• Tossing coins

The main difference between theoretical and experimental probability is that theoretical probability expresses how likely an event is to occur, while experimental probability characterizes how frequently an event occurs in an experiment.

In general, the theoretical probability is more reliable than experimental because it doesn’t rely on a limited sample size; however, experimental probability can still give you a good idea of the chances of something happening.

The reason is that the theoretical probability of an event will invariably be the same, whereas the experimental probability is typically affected by chance; therefore, it can be different for different experiments.

Also, generally, the more trials you carry out, the more times you flip a coin, and the closer the experimental probability is likely to be to its theoretical probability.

Also, note that theoretical probability is calculated using mathematical formulas, while experimental probability is found by conducting experiments.

What to read next:

• Types of Statistics in Mathematics And Their Applications .
• Is Statistics Harder Than Algebra? (Let’s find out!)
• Should You Take Statistics or Calculus in High School?
• Is Statistics Hard in High School? (Yes, here’s why!)

Wrapping Up

Theoretical and experimental probabilities are two ways of calculating the likelihood of an event occurring. Theoretical probability uses mathematical formulas, while experimental probability uses data from experiments. Both types of probability are useful in different situations.

I believe that both theoretical and experimental probabilities are important in mathematics. Theoretical probability uses mathematical formulas to calculate chances, while experimental probability relies on results from experiments or surveys.

I am Altiné. I am the guy behind mathodics.com. When I am not teaching math, you can find me reading, running, biking, or doing anything that allows me to enjoy nature's beauty. I hope you find what you are looking for while visiting mathodics.com.

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• → Curriculum
• → 7th Grade
• → Unit 9: Probability

Theoretical and Experimental Probability Lesson Plan

Get the lesson materials.

Theoretical and Experimental Probability Guided Notes w/ Doodles | Sketch Notes

Ever wondered how to teach theoretical and experimental probability in an engaging way to your 7th grade students?

In this lesson plan, students will learn about probability concepts and their real-life applications. Through artistic and interactive guided notes, check for understanding activities, a doodle and color by number worksheet, and a maze worksheet, students will gain a comprehensive understanding of theoretical and experimental probability.

The lesson concludes with a real-life example that explores how probability can be applied in everyday situations.

• Standards : CCSS 7.SP.A.1 , CCSS 7.SP.A.2 , CCSS 7.SP.C.5 , CCSS 7.SP.C.6 , CCSS 7.SP.C.7 , CCSS 7.SP.C.7.a , CCSS 7.SP.C.7.b
• Topic : Statistics & Probability
• Grade : 7th Grade
• Type : Lesson Plans

Learning Objectives

After this lesson, students will be able to:

Define theoretical probability and experimental probability

Calculate theoretical probability of a simple event

Collect data to approximate experimental probability

Compare and contrast theoretical probability and experimental probability

Apply theoretical and experimental probability to real-life scenarios

Solve problems involving theoretical and experimental probability

Prerequisites

Before this lesson, students should be familiar with:

Basic understanding of probability vocabulary such as outcome, event, sample space, and probability.

Basic understanding of fractions and decimals.

Basic knowledge of data representation such as tables, charts, and graphs.

Colored pencils or markers

Theoretical and Experimental Probability Guided Notes

Key Vocabulary

Theoretical probability

Experimental probability

Chance event

Data collection

Probability

Probability model

Sample space

Introduction

As a hook, ask students why understanding probability is important in everyday life. For example, you can ask them why it's important to know the probability of winning a lottery or the probability of getting a certain outcome in a game. Refer to the last page of the guided notes as well as the FAQs below for more ideas on how to engage students in the discussion.

Use the first page of the guided notes to introduce the concept of theoretical probability vs. experimental probability. Walk through the key points of the topic, including how to calculate the theoretical probability of an event. Emphasize that theoretical probability is based on what we expect to happen in an ideal situation. Explain that experimental probability is based on collecting data from actual trials or experiments. Walk through the key points of the topic, including how to calculate the experimental probability of an event. Emphasize that experimental probability can vary from the theoretical probability because it is based on actual data. Refer to the FAQ below for a walk-through on this, as well as ideas on how to respond to common student questions.

Based on student responses, reteach any concepts that students need extra help with. If your class has a wide range of proficiency levels, you can pull out students for reteaching, and have more advanced students begin work on the practice exercises.

Have students practice theoretical and experimental probability using the practice worksheet activity (pg. 2 of guided notes). Walk around to answer student questions.

Fast finishers can dive into the maze (pg. 3 of guided notes) and color by number activities (pg. 4 of guided notes) on the practice worksheet for extra practice. You can assign it as homework for the remainder of the class.

Real-Life Application

Use the last page of the guided notes (pg. 5) to bring the class back together, and introduce the concept of real-life applications of probability. Explain to students that probability is used in many different fields and industries to help make informed decisions and predictions. Some examples of real-life applications include:

Weather Forecasting: Meteorologists use probability to predict the likelihood of rain, snow, or severe weather. By analyzing historical weather data and current atmospheric conditions, they can calculate the probability of certain weather events occurring in a specific area.

Sports Statistics: Probability is used in sports to analyze player performance, predict game outcomes, and determine the chances of a team making it to the playoffs. Statistics such as shooting percentage in basketball, batting average in baseball, or completion percentage in football are all based on probability.

Insurance: Insurance companies use probability to determine the likelihood of certain events, such as accidents or natural disasters, occurring to their policyholders. This helps them calculate insurance premiums that accurately reflect the level of risk involved.

Refer to the FAQ section in the teaching resource for more ideas on how to incorporate real-life applications of probability into your lessons.

Additional Print Practice

A fun, no-prep way to practice theoretical and experimental probability is Doodle Math. It's a fresh take on color by number or color by code. It includes multiple levels of practice, perfect for a review day or sub plan.

Here are some activities to try:

Theoretical and Experimental Probability | Doodle Math: Twist on Color by Number Worksheets

Additional Self-Checking Digital Practice

If you’re looking for digital practice for theoretical and experimental probability, try my Pixel Art activities in Google Sheets. Every answer is automatically checked, and correct answers unlock parts of a mystery picture. It’s incredibly fun, and a powerful tool for differentiation.

Here are some activities to explore:

Theoretical & Experimental Probability Digital Pixel Art Activities

What is theoretical probability? Open

Theoretical probability is the probability based on mathematical calculations and reasoning, without any actual experimentation or data collection.

How is theoretical probability calculated? Open

Theoretical probability is calculated by dividing the number of desired outcomes by the total number of possible outcomes.

Steps to calculate theoretical probability:

Identify the desired outcomes.

Determine the total number of possible outcomes.

Divide the number of desired outcomes by the total number of possible outcomes.

What is experimental probability? Open

Experimental probability is the probability based on actual data collected from experiments or observations.

How is experimental probability calculated? Open

Experimental probability is calculated by dividing the number of times an event occurs by the total number of trials or observations.

Steps to calculate experimental probability:

Conduct trials or observations.

Count the number of times the event of interest occurs.

Divide the number of occurrences by the total number of trials or observations.

What is the difference between theoretical and experimental probability? Open

The main difference between theoretical and experimental probability is that theoretical probability is based on mathematical calculations and reasoning, while experimental probability is based on actual data collected from experiments or observations.

How can I use theoretical probability in real-life situations? Open

Theoretical probability can be used in real-life situations to make predictions or informed decisions based on mathematical calculations and reasoning. For example:

Predicting the chances of winning a game or lottery.

Determining the likelihood of certain weather conditions.

Estimating the probability of a certain event occurring in a scientific experiment.

How can I use experimental probability in real-life situations? Open

Experimental probability can be used in real-life situations to make predictions or draw conclusions based on actual data collected from experiments or observations. For example:

Estimating the success rate of a new medical treatment based on clinical trials.

Evaluating the effectiveness of a marketing campaign based on customer responses.

Assessing the probability of a certain outcome in sports based on past performance data.

How can I teach theoretical and experimental probability effectively? Open

To teach theoretical and experimental probability effectively, consider the following strategies:

Use visual aids and manipulatives to illustrate the concept.

Provide real-life examples and applications to make the topic relatable.

Engage students in hands-on activities or experiments to collect data and calculate probabilities.

Incorporate interactive discussions and problem-solving tasks to encourage critical thinking.

Offer opportunities for students to practice and apply their knowledge through worksheets or interactive games.

Want more ideas and freebies?

Get my free resource library with digital & print activities—plus tips over email.

Theoretical Probability with Dice Practice Grid ( Editable Word | PDF | Answers )

Theoretical Probability with Spinners Practice Grid ( Editable Word | PDF | Answers )

Theoretical Probability with Counters Practice Grid ( Editable Word | PDF | Answers )

Theoretical Probability with Playing Cards Practice Grid ( Editable Word | PDF | Answers )

Theoretical Probability Practice Strips ( Editable Word | PDF | Answers )

Theoretical Probability Odd One Out ( Editable Word | PDF | Answers )

Constructing Two-Way Tables Practice Grid ( Editable Word | PDF | Answers )

Completing Two-Way Tables Practice Grid ( Editable Word | PDF | Answers )

Finding Probability from Two-Way Tables Practice Grid ( Editable Word | PDF | Answers )

Two-Way Tables and Probability Practice Strips ( Editable Word | PDF | Answers )

Experimental Probability Practice Strips ( Editable Word | PDF | Answers )

Estimating Probability Experiments Activity ( Editable Word | PDF )

Theoretical and Experimental Probability Revision Practice Grid ( Editable Word | PDF | Answers ​ )

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Probability

• Leaderboard
• Introduction to Probability:  Probability is the likelihood of an event happening. It's often represented as a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain. 
• Probability of Simple Events:  For simple events, where each outcome is equally likely, the probability of an event happening is calculated by dividing the number of favourable outcomes by the total number of possible outcomes.  
• Probability of Compound Events:  For compound events, where multiple events are happening together, the probability is calculated using methods like the " AND " rule (multiplying probabilities) or the " OR" rule (adding probabilities).
•  Probability as Fractions, Decimals, and Percentages:  Probability can be expressed as fractions, decimals, or percentages. For example, a probability of 1/4 is the same as 0.25 or 25%.
•  Experimental Probability vs. Theoretical Probability:  Experimental probability is based on actual experiments or observations , while theoretical probability is calculated based on mathematical principles and assumptions.
•  Independent and Dependent Events:  Events are independent if the outcome of one event doesn't affect the outcome of another. Events are dependent if the outcome of one event does affect the outcome of another.   HINT: IF THE OUTCOME DOES NOT AFFECT ANOTHER OUTCOME IT IS INDEPENDENT (ON ITS OWN)
• Tree Diagrams and Probability Tables:  Tree diagrams and probability tables are tools used to visualise and organise the outcomes of multiple events , making it easier to calculate probabilities. Applications of Probability: Probability is used in various real-life situations, such as weather forecasting, sports predictions, and gambling. 

• Find Gizmos
• Lesson Info

Spin the Big Wheel! (Probability)

Step right up! Spin the big wheel! Each spin can result in no prize, a small prize, or a big prize. The wheel can be spun by 1, 10, or 100 players. Results are recorded on a frequency table or a circle graph. You can also design your own wheel and a sign that describes the probabilities for your wheel.

Learning Objectives

• Distinguish between events that are certain, likely, unlikely, and impossible.
• Understand that probability refers to the likelihood of an event, but does not allow the event to be predicted with certainty.
• Determine the probability of a given outcome, such as winning a prize with a spinner.
• Understand that experimental results will not exactly match theoretical probability.

certain, impossible, outcome, probability, sample space, trial

Lesson Materials

Student Exploration Sheet

Exploration Sheet Answer Key

Assessment Questions

Teacher Guide

Vocabulary Sheet

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