What is an example of theoretical probability?

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The following table highlights the difference between Experimental Probability and Theoretical Probability. Scroll down the page for more examples and solutions.

How To Find The Experimental Probability Of An Event?

Step 1: Conduct an experiment and record the number of times the event occurs and the number of times the activity is performed.

Step 2: Divide the two numbers to obtain the Experimental Probability.

How To Find The Theoretical Probability Of An Event?

The Theoretical Probability of an event is the number of ways the event can occur (favorable outcomes) divided by the number of total outcomes.

What Is The Theoretical Probability Formula?

The formula for theoretical probability of an event is

Experimental Probability

One way to find the probability of an event is to conduct an experiment.

Example:
A bag contains 10 red marbles, 8 blue marbles and 2 yellow marbles. Find the experimental probability of getting a blue marble.

Solution: Take a marble from the bag. Record the color and return the marble. Repeat a few times (maybe 10 times).

Count the number of times a blue marble was picked (Suppose it is 6).

The experimental probability of getting a blue marble from the bag is

How to find and use experimental probability?

The following video gives another example of experimental probability.

• Show Video Lesson

How the results of the experimental probability may approach the theoretical probability?

Example: The spinner below shows 10 equally sized slices. Heather spun 50 times and got the following results. a) From Heather’s' results, compute the experimental probability of landing on yellow.

b) Assuming that the spinner is fair, compute the theoretical probability of landing in yellow.

• Show Video Lesson

Theoretical Probability

We can also find the theoretical probability of an event.

Example:
A bag contains 10 red marbles, 8 blue marbles and 2 yellow marbles. Find the theoretical probability of getting a blue marble.

Solution: There are 8 blue marbles. Therefore, the number of favorable outcomes = 8.

There are a total of 20 marbles. Therefore, the number of total outcomes = 20

Example:
Find the probability of rolling an even number when you roll a die containing the numbers 1-6. Express the probability as a fraction, decimal, ratio and percent.

Solution: The possible even numbers are 2, 4, 6. Number of favorable outcomes = 3.

Total number of outcomes = 6

The probability =

Comparing Theoretical And Experimental Probability

The following video gives an example of theoretical and experimental probability.

Example: According to theoretical probability, how many times can we expect to land on each color in a spinner, if we take 16 spins?

Conduct the experiment to get the experimental probability.

We will then compare the Theoretical Probability and the Experimental Probability.

• Show Video Lesson

The following video shows another example of how to find the theoretical probability of an event.

Examples:

1. A spinner is divided into eight equal sectors, numbered 1 through 8. a) What is the probability of spinning an odd numbers? b) What is the probability of spinning a number divisible by 4?

   b) What is the probability of spinning a number less than 3?

2. A spinner is divided into eight equal sectors, numbered 1 through 8. a) What is the probability of spinning a 2? b) What is the probability of spinning a number from 1 to 4?

   b) What is the probability of spinning a number divisible by 2?

• Show Video Lesson

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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:

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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 \(\frac{Number\: of\: favorable \:outcomes}{Number\: of \:possible \:outcomes}\).

1. 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.

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

   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.

2. 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.

3. 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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FAQs on Theoretical Probability

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.