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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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Theoretical Probability

Theoretical probability as the name suggests is the theory behind probability. Theoretical probability gives the outcome of the occurrence of an event based on mathematics and reasoning. It tells us about what should happen in an ideal situation without conducting any experiments.

Theoretical probability is extremely useful in situations, such as in the launching of a satellite, where it is not feasible to conduct an actual experiment to arrive at a sound conclusion. In this article, we will learn more about the meaning of theoretical probability, the differences between the types of probabilities, and see some associated examples.

What is Theoretical Probability?

Theoretical probability is an approach in probability theory that is used to calculate the probability of an outcome of a specific event. Probability theory is a branch of mathematics that is concerned with finding the likelihood of occurrence of a random event. The probability that an event will occur lies between 0 and 1. If the probability is closer to 0 it implies that the event is less likely to take place. Similarly, if the probability is closer to 1 it denotes that the event has a higher chance of occurring.

Theoretical Probability Definition

Theoretical probability can be defined as the number of favorable outcomes divided by the total number of possible outcomes. To determine the theoretical probability there is no need to conduct an experiment. However, knowledge of the situation is required to find the probability of occurrence of that event. Theoretical probability predicts the probability of occurrence of an event by assuming that all events are equally likely to occur.

Theoretical Probability Example

Suppose there are a total of 5 cards and the probability of drawing 2 cards needs to be determined. Then by using the concept of theoretical probability, the number of favorable outcomes (2) is divided by the total possible outcomes (5) to get the probability as 0.4.

Theoretical Probability Formula

Theoretical probability can be calculated either by using logical reasoning or by using a simple formula. The result of such a type of probability is based on the number of possible outcomes. The theoretical probability formula is equal to the ratio of the number of favorable outcomes to the total number of probable outcomes. This formula is expressed as follows:

Theoretical Probability = Number of favorable outcomes / Number of possible outcomes.

How to Find Theoretical Probability?

Theoretical probability is used to express the likelihood of occurrence of an event without conducting any experiments. Suppose a person has 30 raffle tickets and, in total, 500 tickets were sold. The steps to calculate the theoretical probability of the person winning a prize are as follows:

• Step 1: Identify the number of favorable outcomes. As there are 30 raffle tickets thus, 30 will be the number of desired outcomes.
• Step 2: Determine the total possible outcomes. Since 500 total tickets were sold thus, 500 will be the number of total possible outcomes.
• Step 3: To calculate the theoretical probability divide the value from step 1 by step 2. Thus, 30 / 500 = 0.06. This shows that the probability of a person winning a raffle prize is 0.06.

Theoretical Probability vs Empirical Probability

Empirical probability is also known as experimental probability . Both theoretical probability and empirical probability are approaches to calculating the chance that a random event will occur. The difference between theoretical probability and empirical probability is given in the table below:

Related Articles:

• Probability Rules
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• Poisson Distribution Formula
• Event Probability Calculator

Important Notes on Theoretical Probability

• Theoretical probability is used to calculate the probability of occurrence of an event without performing an experiment.
• Theoretical probability assumes that all events have an equal likelihood of occurrence.
• The theoretical probability formula is given as \(\frac{Number\: of\: favorable \:outcomes}{Number\: of \:possible \:outcomes}\).

Examples on Theoretical Probability

Example 1: If a bag contains 5 red and 7 blue balls then what is the probability of picking up a red ball?

Solution: To calculate the theoretical probability the following formula is used.

Number of favorable outcomes = 5

Number of possible outcomes = 5 + 7 = 12

P(red) = 5 / 12 = 0.4167

Answer: The probability of picking up a red ball is 0.4167.

Example 2: Find the probability of getting 3 on a fair die.

Solution: The possible outcomes of rolling a die are 1, 2, 3, 4, 5, 6.

In other words, the total number of outcomes = 6

As we want to calculate the probability of getting only number 3 thus, the number of favorable outcomes = 1

P(3) = 1 / 6 = 0.167

Answer: The probability of getting 3 on a fair die is 0.167.

Example 3: The letters of the word "MATHEMATICS" are placed in a bag. What is the probability of drawing the letter "T" from the bag?

Solution: The total number of letters = 11

As there are two T's placed in the bag thus, the number of favorable outcomes = 2.

P(T) = 2 / 11 = 0.182

Answer: The probability of drawing the letter "T" is 0.182

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Practice Questions on Theoretical Probability

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FAQs on Theoretical Probability

What is theoretical probability in math.

Theoretical probability in math refers to the probability that is calculated without any experiment being performed. It can be defined as the ratio of the number of favorable outcomes to the total number of possible outcomes.

What is the Formula for Theoretical Probability?

The theoretical probability formula is given as follows:

Theoretical probability = \(\frac{Number\: of\: favorable \:outcomes}{Number\: of \:possible \:outcomes}\)

How Do You Calculate Theoretical Probability?

To calculate the theoretical probability the steps are as follows:

• Find the number of desired or favorable outcomes.
• Find the total number of outcomes.
• Divide the value obtained in step 1 by the value from step 2.

What is the Difference Between Theoretical Probability and Experimental Probability?

Theoretical probability is calculated when conducting an experiment is not possible. It gives a fair idea of the likelihood of occurrence of an outcome. In contrast, experimental probability is calculated based on experiments that have been conducted in the past.

Is Classical Probability the Same as Theoretical Probability?

Yes, theoretical probability is the same as classical probability. It is an approach used to calculate the outcome of an event based on the assumption that each outcome of the given event is equally likely to occur.

Why Do You Use Theoretical Probability?

Theoretical probability is required in situations where it is not possible to conduct repeated experiments due to its impracticality or lack of finance. For example, direct experiments cannot be performed when determining the various probabilities associated with subatomic particles. In such a case theoretical probability is used.

What is a Theoretical Probability Distribution?

Theoretical distributions that are designed based on certain assumptions are known as theoretical probability distributions. They are required for the quick analysis of the distribution of random variables. For example, Bernoulli distribution , normal distribution , binomial distribution , etc. are theoretical probability distributions.

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Home / United States / Math Classes / 7th Grade Math / Experimental and Theoretical Probability

Experimental and Theoretical Probability

Probability is a branch of math that studies the chance or likelihood of an event occurring. There are two types of prob ability for a particular event: experimental probability and theoretical probability. Learn the difference between the two types of probabilities and the steps involved in their calculation. ...Read More Read Less

Experimental and Theoretical Probability in Math

What is Probability?

• Experimental Probability
• Theoretical Probability
• Solved Examples
• Frequently Asked Questions

Th e chance of a happening is named as the probability of the event happening. It tells us how likely an occasion is going to happen; it doesn’t tell us what’s happening. There is a fair chance of it happening (happening/not happening). They’ll be written as decimals or fractions . The probability of occurrence A is below.

P (A) = \(\frac{\text{Number of favorable outcomes}}{\text{Number of total possible outcomes}}\)

Following are two varieties of probability:

• Experimental probability
• Theoretical probability

What is Experimental Probability

Definition : Probability that’s supported by repeated trials of an experiment is named as experimental probability.

P (event) = \(\frac{\text{Number of times that event occurs}}{\text{Total number of trails}}\)

Example: The table shows the results of spinning a penny 62 times. What’s the probability of spinning heads?

Solution: Heads were spun 23 times in a total of 23 + 39 = 62 spins.

P (heads) = \(\frac{\text{23}}{\text{69}}\) = 0.37  or 37.09 %

What is Theoretical Probability

Definition : When all possible outcomes are equally likely the theoretical possibility of an incident is that the quotient of the number of favorable outcomes and therefore the number of possible outcomes.

P (event) = \(\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}\)

Example: You randomly choose one among the letters shown. What’s the theoretical probability of randomly choosing an X?

Solution: P (x) = \(\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}\) = \(\frac{\text{1}}{\text{7}}\) or 14.28%

A prediction could be a reasonable guess about what is going to happen in the future. Good predictions should be supported by facts and probability.

Predictions supported theoretical probability. These are the foremost reliable varieties of predictions, based on physical relationships that are easy to work and measure which don’t change over time. They include such things as:

• number cubes

Let’s take a look at some differences between experimental and theoretical probability:

Theoretical & Experimental Probability Examples

1. What is the probability of tossing a variety cube and having it come up as a two or a three?

Solution:  

First, find the full number of outcomes

Outcomes: 1, 2, 3, 4, 5, and 6

Total outcomes = 6

Next, find the quantity of favorable outcomes.

Favorable outcomes:

Getting a 2 or a 3 = 2 favorable outcomes

Then, find the ratio of favorable outcomes to total outcomes.

P (Event) = Number of favorable outcomes : total number of outcomes

P (2 or 3) = 2:6

P (2 or 3) = 1:3

The solution is 1:3

The theoretical probability of rolling a 2 or a 3 on a variety of cube is 1:3.

2 . A bag contains 25 marbles. You randomly draw a marble from the bag, record its color, so replace it. The table shows the results after 11 draws. Predict the amount of red marbles within the bag.

To seek out the experimental probability of drawing a red marble.

P (EVENT) = \(\frac{\text{Number of times the event occurs}}{\text{Total number of trials}}\)

P (RED) = \(\frac{\text{5}}{\text{11}}\)        (You draw red 5 times. You draw a complete of 11 marbles)

To make a prediction, multiply the probability of drawing red by the overall number of marbles within the bag.

\(\frac{\text{5}}{\text{11}}\) x 25 = 11.36 ~ 11 so you’ll be able to predict that there are 11 red balls in an exceedingly bag

3. A spinner was spun 1000 times and the frequency of outcomes was recorded as in the given table.

Find (a) list the possible outcomes that you can see in the spinner (b) compare the probability of each outcome (c) find the ratio of each outcome to the total number of times that the spinner spun.

(a) T he possible outcomes are 5. They are red, orange, purple, yellow, and green. Here all the five colors occupy the same area in the spinner. They are all equally likely.

(b) Compute the probability of each event.

P (Red) = \(\frac{\text{Favorable outcomes of red}}{\text{Total number of possible outcomes}}\) = \(\frac{\text{1}}{\text{5}}\) = 0.2

Similarly, P (Orange), P (Purple), P (Yellow) and P (Green) are also \(\frac{\text{1}}{\text{5}}\) or 0.2.

(c) From the experiment the frequency was recorded in the table.

Ratio for red = \(\frac{\text{Number of outcomes of red in the above experiment}}{\text{Number of times the spinner was spun}}\) = \(\frac{\text{185}}{\text{1000}}\) = 0.185

Similarly, we can find the corresponding ratios for orange, purple, yellow, and green are 0.195, 0.210, 0.206, and 0.204 respectively. Can you see that each of the ratios is approximately equal to the probability which we have obtained in (b) [i.e. before conducting the experiment]

How do you find experimental probability?

The experimental probability of an occurrence is predicted by actual experiments and therefore the recordings of the events. It’s adequate to the amount of times an incident occurred divided by the overall number of trials.

How does one find theoretical probability?

When all possible events or outcomes are equally likely to occur, the theoretical probability is found without collecting data from an experiment.

What is experimental probability used for?

Experimental probability, also called Empirical probability, relies on actual experiments and adequate recordings of the happening of events. To work out the occurrence of any event, a series of actual experiments are conducted.

Why is experimental probability different from theoretical?

Theoretical probability describes how likely an occurrence is to occur. We all know that a coin is equally likely to land heads or tails, therefore the theoretical probability of getting heads is 1/2. Experimental probability describes how frequently a happening actually occurred in an experiment.

Is flipping a coin theoretical or experimental probability?

So the results of flipping a coin should be somewhere around 50% heads and 50% tails since that’s the theoretical probability.

Can the experimental probability of an incident be a negative number?

No, since the quantity of trials during which the event can happen can not be negative and also the total number of trials is usually positive.

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