Statistics Made Easy
Theoretical Probability: Definition + Examples
Probability is a topic in statistics that describes the likelihood of certain events happening. When we talk about probability, we’re often referring to one of two types:
1. Theoretical probability
Theoretical probability is the likelihood that an event will happen based on pure mathematics. The formula to calculate the theoretical probability of event A happening is:
P( A ) = number of desired outcomes / total number of possible outcomes
For example, the theoretical probability that a dice lands on “2” after one roll can be calculated as:
P( land on 2 ) = (only one way the dice can land on 2) / (six possible sides the dice can land on) = 1/6
2. Experimental probability
Experimental probability is the actual probability of an event occurring that you directly observe in an experiment. The formula to calculate the experimental probability of event A happening is:
P( A ) = number of times event occurs / total number of trials
For example, suppose we roll a dice 11 times and it lands on a “2” three times. The experimental probability for the dice landing on “2” can be calculated as:
P( land on 2 ) = (lands on 2 three times) / (rolled the dice 11 times) = 3/11
How to Remember the Difference
You can remember the difference between theoretical probability and experimental probability using the following trick:
- The theoretical probability of an event occurring can be calculated in theory using math.
- The experimental probability of an event occurring can be calculated by directly observing the results of an experiment .
The Benefit of Using Theoretical Probability
Statisticians often like to calculate the theoretical probability of events because it’s much easier and faster to calculate compared to actually conducting an experiment.
For example, suppose it’s known that 1 out of every 30 students at a particular school will need additional help with their math homework after school. Instead of waiting to see how many students show up for homework help after school, a school administrator could instead calculate the total number of students at the school (suppose it’s 300) and multiply by the theoretical probability (1/30) to know that he will likely need 10 people present to help each of the students one-on-one.
Examples of Theoretical Probability
Experimental probabilities are usually easier to calculate than theoretical probabilities because it just involves counting the number of times that a certain event actually occurred relative to the total number of trials.
Conversely, theoretical probabilities can be trickier to calculate. So, here are several examples of how to calculate theoretical probabilities to help you master the topic.
A bag contains the following:
- 3 red balls
- 4 green balls
- 2 purple balls
Question: If you close your eyes and randomly pull out one ball, what is the probability that it will be green?
Answer: We can use the following formula to calculate the theoretical probability of pulling out a green ball:
P( green ) = (4 green balls) / (9 total balls) = 4/9
You own a 9-sided dice that contains the numbers 1 through 9 on the sides.
Question: What is the probability that the dice lands on “7” if you were to roll it one time?
Answer: We can use the following formula to calculate the theoretical probability that the dice lands on 7:
P( lands on 7 ) = (only one way the dice can land on 7) / (9 possible sides) = 1/9
A bag contains the name of 3 boys and 7 seven girls.
Question: If you close your eyes and randomly pull one name out of the bag, what is the probability that you pull out a girl’s name?
Answer: We can use the following formula to calculate the theoretical probability that you pull out a girl’s name:
P( girls name ) = (7 possible girl names) / (10 total names) = 7/10
Published by Zach
Leave a reply cancel reply.
Your email address will not be published. Required fields are marked *
3.6 Probability Topics
Probability topics.
Class time:
Student Learning Outcomes
- The student will use theoretical and empirical methods to estimate probabilities.
- The student will appraise the differences between the two estimates.
- The student will demonstrate an understanding of long-term relative frequencies.
Do the Experiment Count out 40 mixed-color M&Ms®, which is approximately one small bag’s worth. Record the number of each color in Table 3.12 . Use the information from this table to complete Table 3.13 .
Next, put the M&Ms in a cup. The experiment is to pick two M&Ms, one at a time. Do not look at them as you pick them. The first time through, replace the first M&M before picking the second one. Record the results in the “With Replacement” column of Table 3.14 . Do this 12 times.
The second time through, after picking the first M&M, do not replace it before picking the second one. Then, pick the second one. Record the results in the “Without Replacement” column section of Table 3.13 . After you record the pick, put both M&Ms back. Do this a total of 12 times, also.
Use the data from Table 3.14 to calculate the empirical probabilities shown in Table 3.15 . Leave your answers in unreduced fractional form. Do not multiply out any fractions.
G 2 = green on second pick; R 1 = red on first pick; B 1 = brown on first pick; B 2 = brown on second pick; doubles = both picks are the same colour.
Discussion Questions
- Why are the “With Replacement” and “Without Replacement” probabilities different?
- Theoretical “With Replacement”: P (no yellows) = _______
- Empirical “With Replacement”: P (no yellows) = _______
- Are the decimal values “close”? Did you expect them to be closer together or farther apart? Why?
- If you increased the number of times you picked two M&Ms to 240 times, why would empirical probability values change?
- Would this change (see part 3) cause the empirical probabilities and theoretical probabilities to be closer together or farther apart? How do you know?
- Explain the differences in what P ( G 1 AND R 2 ) and P ( R 1 | G 2 ) represent. Hint: Think about the sample space for each probability.
This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.
Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.
Access for free at https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
- Authors: Barbara Illowsky, Susan Dean
- Publisher/website: OpenStax
- Book title: Introductory Statistics 2e
- Publication date: Dec 13, 2023
- Location: Houston, Texas
- Book URL: https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
- Section URL: https://openstax.org/books/introductory-statistics-2e/pages/3-6-probability-topics
© Dec 6, 2023 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.
Simple and Theoretical Probability
Popular tutorials in simple and theoretical probability.
How Do You Find the Probability of a Simple Event?
Working with probabilities? Check out this tutorial! You'll see how to calculate the probability of picking a certain marble out of a bag.
How Do You Find the Probability of the Complement of an Event?
Probability can help you solve all sorts of everyday problems! This tutorial shows you how to find the probability of the complement of an event using gummy worms!
What is an Outcome?
When you're conducting an experiment, the outcome is a very important part. The outcome of an experiment is any possible result of the experiment. Learn about outcomes by watching this tutorial!
What is a Sample Space?
In an experiment, it's good to know your sample space. The sample space is the set of all possible outcomes of an experiment. Watch this tutorial to get a look at the sample space of an experiment!
What is Probability?
Probability can help you solve all sorts of everyday problems, but first you need to know what probability is! Follow along with this tutorial to learn about probability!
What is the Complement of an Event?
When you learn about probablilities, the complement of an event is a must-know term! This tutorial introduces you the complement of an event.
Related Topics
Other topics in theoretical and experimental probability :.
- Experimental and Geometric Probability
- Terms of Use
Theoretical Probability: Definition + Examples
Probability is a topic in statistics that describes the likelihood of certain events happening. When we talk about probability, we’re often referring to one of two types:
1. Theoretical probability
Theoretical probability is the likelihood that an event will happen based on pure mathematics. The formula to calculate the theoretical probability of event A happening is:
P( A ) = number of desired outcomes / total number of possible outcomes
For example, the theoretical probability that a dice lands on “2” after one roll can be calculated as:
P( land on 2 ) = (only one way the dice can land on 2) / (six possible sides the dice can land on) = 1/6
2. Experimental probability
Experimental probability is the actual probability of an event occurring that you directly observe in an experiment. The formula to calculate the experimental probability of event A happening is:
P( A ) = number of times event occurs / total number of trials
For example, suppose we roll a dice 11 times and it lands on a “2” three times. The experimental probability for the dice landing on “2” can be calculated as:
P( land on 2 ) = (lands on 2 three times) / (rolled the dice 11 times) = 3/11
How to Remember the Difference
You can remember the difference between theoretical probability and experimental probability using the following trick:
- The theoretical probability of an event occurring can be calculated in theory using math.
- The experimental probability of an event occurring can be calculated by directly observing the results of an experiment .
The Benefit of Using Theoretical Probability
Statisticians often like to calculate the theoretical probability of events because it’s much easier and faster to calculate compared to actually conducting an experiment.
For example, suppose it’s known that 1 out of every 30 students at a particular school will need additional help with their math homework after school. Instead of waiting to see how many students show up for homework help after school, a school administrator could instead calculate the total number of students at the school (suppose it’s 300) and multiply by the theoretical probability (1/30) to know that he will likely need 10 people present to help each of the students one-on-one.
Examples of Theoretical Probability
Experimental probabilities are usually easier to calculate than theoretical probabilities because it just involves counting the number of times that a certain event actually occurred relative to the total number of trials.
Conversely, theoretical probabilities can be trickier to calculate. So, here are several examples of how to calculate theoretical probabilities to help you master the topic.
A bag contains the following:
- 3 red balls
- 4 green balls
- 2 purple balls
Question: If you close your eyes and randomly pull out one ball, what is the probability that it will be green?
Answer: We can use the following formula to calculate the theoretical probability of pulling out a green ball:
P( green ) = (4 green balls) / (9 total balls) = 4/9
You own a 9-sided dice that contains the numbers 1 through 9 on the sides.
Question: What is the probability that the dice lands on “7” if you were to roll it one time?
Answer: We can use the following formula to calculate the theoretical probability that the dice lands on 7:
P( lands on 7 ) = (only one way the dice can land on 7) / (9 possible sides) = 1/9
A bag contains the name of 3 boys and 7 seven girls.
Question: If you close your eyes and randomly pull one name out of the bag, what is the probability that you pull out a girl’s name?
Answer: We can use the following formula to calculate the theoretical probability that you pull out a girl’s name:
P( girls name ) = (7 possible girl names) / (10 total names) = 7/10
Explanatory & Response Variables: Definition & Examples
Related posts, how to normalize data between -1 and 1, how to create a vector of ones in..., vba: how to check if string contains another..., how to interpret f-values in a two-way anova, how to determine if a probability distribution is..., what is a symmetric histogram (definition & examples), how to find the mode of a histogram..., how to find quartiles in even and odd..., how to calculate sxy in statistics (with example), how to calculate sxx in statistics (with example).
If you're seeing this message, it means we're having trouble loading external resources on our website.
If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.
To log in and use all the features of Khan Academy, please enable JavaScript in your browser.
Course: 7th grade > Unit 7
- Statistics and probability FAQ
Intro to theoretical probability
- Simple probability: yellow marble
- Simple probability: non-blue marble
- Simple probability
- Experimental probability
- Intuitive sense of probabilities
- Comparing probabilities
Want to join the conversation?
- Upvote Button navigates to signup page
- Downvote Button navigates to signup page
- Flag Button navigates to signup page
Video transcript
- school Campus Bookshelves
- menu_book Bookshelves
- perm_media Learning Objects
- login Login
- how_to_reg Request Instructor Account
- hub Instructor Commons
- Download Page (PDF)
- Download Full Book (PDF)
- Periodic Table
- Physics Constants
- Scientific Calculator
- Reference & Cite
- Tools expand_more
- Readability
selected template will load here
This action is not available.
4.1: Empirical Probability
- Last updated
- Save as PDF
- Page ID 130244
- Kathryn Kozak
- Coconino Community College
One story about how probability theory was developed is that a gambler wanted to know when to bet more and when to bet less. He talked to a couple of friends of his that happened to be mathematicians. Their names were Pierre de Fermat and Blaise Pascal. Since then many other mathematicians have worked to develop probability theory.
Understanding probabilities are important in life. Examples of mundane questions that probability can answer for you are if you need to carry an umbrella or wear a heavy coat on a given day. More important questions that probability can help with are your chances that the car you are buying will need more maintenance, your chances of passing a class, your chances of winning the lottery, your chances of being in a car accident, and the chances that the U.S. will be attacked by terrorists. Most people do not have a very good understanding of probability, so they worry about being attacked by a terrorist but not about being in a car accident. The probability of being in a terrorist attack is much smaller than the probability of being in a car accident, thus it actually would make more sense to worry about driving. Also, the chance of you winning the lottery is very small, yet many people will spend the money on lottery tickets. Yet, if instead they saved the money that they spend on the lottery, they would have more money. In general, events that have a low probability (under 5%) are unlikely to occur. Whereas if an event has a high probability of happening (over 80%), then there is a good chance that the event will happen. This chapter will present some of the theory that you need to help make a determination of whether an event is likely to happen or not.
First you need some definitions.
Definition \(\PageIndex{1}\)
Experiment : an activity that has specific result that can occur, but it is unknown which results will occur.
Definition \(\PageIndex{2}\)
Outcomes : the result of an experiment.
Definition \(\PageIndex{3}\)
Event : a set of certain outcomes of an experiment that you want to have happen.
Definition \(\PageIndex{4}\)
Sample Space : collection of all possible outcomes of the experiment. Usually denoted as SS.
Definition \(\PageIndex{5}\)
Event Space : the set of outcomes that make up an event. The symbol is usually a capital letter.
Start with an experiment. Suppose that the experiment is rolling a die. The sample space is {1, 2, 3, 4, 5, 6}. The event that you want is to get a 6, and the event space is {6}. To do this, roll a die 10 times. When you do that, you get a 6 two times. Based on this experiment, the probability of getting a 6 is 2 out of 10 or 1/5. To get more accuracy, repeat the experiment more times. It is easiest to put this in a table, where n represents the number of times the experiment is repeated. When you put the number of 6s found over the number of times you repeat the experiment, this is the relative frequency.
Notice that as n increased, the relative frequency seems to approach a number. It looks like it is approaching 0.163. You can say that the probability of getting a 6 is approximately 0.163. If you want more accuracy, then increase n even more.
These probabilities are called experimental probabilities since they are found by actually doing the experiment. They come about from the relative frequencies and give an approximation of the true probability. The approximate probability of an event A , P(A) , is
Definition \(\PageIndex{6}\)
Experimental Probabilities
\(P(A)=\dfrac{\text { number of times } A \text { occurs }}{\text { number of times the experiment was repeated }}\)
For the event of getting a 6, the probability would by \(\dfrac{163}{1000}=0.163\).
You must do experimental probabilities whenever it is not possible to calculate probabilities using other means. An example is if you want to find the probability that a family has 5 children, you would have to actually look at many families, and count how many have 5 children. Then you could calculate the probability. Another example is if you want to figure out if a die is fair. You would have to roll the die many times and count how often each side comes up. Make sure you repeat an experiment many times, because otherwise you will not be able to estimate the true probability. This is due to the law of large numbers.
Definition \(\PageIndex{7}\)
Law of large numbers : as n increases, the relative frequency tends towards the actual probability value.
Probability, relative frequency, percentage, and proportion are all different words for the same concept. Also, probabilities can be given as percentages, decimals, or fractions.
Exercise \(\PageIndex{1}\)
- In Australia in 1995, of the 2907 indigenous people in prison 17 of them died. In that same year, of the 14501 non-indigenous people in prison 42 of them died ("Aboriginal deaths in," 2013). Find the probability that an indigenous person dies in prison and the probability that a non-indigenous person dies in prison. Compare these numbers and discuss what the numbers may mean.
1. P(blue) = 0.184, P(brow) = 0.142, P(green) = 0.184, P(orange) = 0.208, P(red) = 0.142, P(yellow) = 0.141
3. P(indigenous person dies) = 0.0058, P(non-indigenous person dies) = 0.0029, see solutions
IMAGES
VIDEO
COMMENTS
If the outcomes are equally likely, then you can do theoretical probabilities. Definition 4.2.1: Theoretical Probabilities. If the outcomes of an experiment are equally likely, then the probability of event A happening is. P(A) = # of outcomes in event space # of outcomes in sample space.
Theoretical probability is the likelihood that an event will happen based on pure mathematics. The formula to calculate the theoretical probability of event A happening is: P (A) = number of desired outcomes / total number of possible outcomes. For example, the theoretical probability that a dice lands on "2" after one roll can be ...
1. A number between I and 3 is chosen 30 times. Results are shown in the table below. Result Frequency 12 2. The spinner below is spun 60 times. Results are shown in the table a) Find and compare the theoretical probability and experimental probability of choosing a 2. Theoretical: 3 Compare: Experimental: SO — hi her- a) Find and compare the ...
The experimental probability of an event is an estimate of the theoretical (or true) probability, based on performing a number of repeated independent trials of an experiment, counting the number of times the desired event occurs, and finally dividing the number of times the event occurs by the number of trials of the experiment. For example, if a fair die is rolled 20 times and the number 6 ...
The theoretical probability approach is used when 𝑆𝑆 consists of a reasonably small number of countable outcomes, each of which is equally likely to occur. There are three steps to calculating a theoretical probability: 1. Define the experiment, the sample space 𝑆𝑆, and the event space 𝐴𝐴. 2.
Probability tells us how often some event will happen after many repeated trials. You've experienced probability when you've flipped a coin, rolled some dice, or looked at a weather forecast. Go deeper with your understanding of probability as you learn about theoretical, experimental, and compound probability, and investigate permutations, combinations, and more!
1.1 Definitions of Statistics, Probability, and Key Terms; 1.2 Data, Sampling, and Variation in Data and Sampling; 1.3 Frequency, ... Homework; Bringing It Together: Homework; References; Solutions; 3 Probability Topics. Introduction; ... The student will use theoretical and empirical methods to estimate probabilities.
Unit 7: Probability. 0/1600 Mastery points. Basic theoretical probability Probability using sample spaces Basic set operations Experimental probability. Randomness, probability, and simulation Addition rule Multiplication rule for independent events Multiplication rule for dependent events Conditional probability and independence.
It can be written as the ratio of the number of favorable events divided by the number of possible events. For example, if you have two raffle tickets and 100 tickets were sold: Ratio = number of favorable outcomes / number of possible outcomes = 2/100 = .5. A theoretical probability distribution is a known distribution like the normal ...
Date: _____ Per: _____ Homework 3: Theoretical vs. Experimental Probability ** This is a 2 Give each probability as a simplified fraction, decimal, and percent. 1. A number between 1 and 3 is chosen 30 times. Results are shown in the table below. a) Find and compare the theoretical probability and experimental probability of choosing a 2 ...
Figure 4-4 shows a graph of experimental probabilities as n gets larger and larger. The dashed yellow line is the theoretical probability of rolling a four of 1/6 \(\neq\) 0.1667. Note the x-axis is in a log scale. Note that the more times you roll the die, the closer the experimental probability gets to the theoretical probability. Figure 4-4
The dashed yellow line is the theoretical probability of rolling a 4, which is \(\dfrac{1}{6}\) \(\approx\) 0.1667. Note the x-axis is in a log scale. Note that the more times you roll the die, the closer the experimental probability gets to the theoretical probability, which illustrates the Law of Large Numbers. Figure \(\PageIndex{2}\)
Probability and Statistics. Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. In this non-linear system, users are free to take whatever path through the material best serves their needs. These unique features make Virtual ...
Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. In this non-linear system, users are free to take whatever path through the material best serves their needs. These unique features make Virtual Nerd a viable alternative to private tutoring.
Theoretical probability is the likelihood that an event will happen based on pure mathematics. The formula to calculate the theoretical probability of event A happening is: P ( A) = number of desired outcomes / total number of possible outcomes. For example, the theoretical probability that a dice lands on "2" after one roll can be ...
First, what's the probability of getting a head per flip? It is 1 / 2 = 0.5. Now, what's the probability of getting 2 heads in 2 flip? You have to flip head for the first time, and for the second time as well, so it will be 0.5 * 0.5 = 0.25. You can expand this to 1000 times, which is 0.5^1000.
Unit 8: Probability & Statistics Homework 2: Simple Probability ** This is a 2-page document! ** Directions: Find each probability as a fraction (in simplest form), decimal, and percent. I. The spinner below is spun once. 12 11 10 2. A ball is thrown into one of the jars. a) p (even) 0.5 , 5070
The probability of two or more independent events is the product of the probabilities of the events. P (A and B) = P (A)*P (B) P (A and B and C) = P (A) P (B) P (C) Probability of Dependent Events. The probability of two or more dependent events A and B is the probability of A times the probability of B after A occurs.
This Probability and Statistics Unit Bundle includes guided notes, homework assignments, two quizzes, a study guide and a unit test that cover the following topics: • The Fundamental Counting Principle. • Permutations. • Combinations. • Theoretical Probability.
In this case, the number of favorable outcomes is 1 (getting heads) and the total number of possible outcomes is 2 (getting either heads or tails). Using the formula, the theoretical probability of flipping heads is: Theoretical Probability = 1 / 2 = 0.5 or 50% So, the theoretical probability of flipping heads is 50%.
Unit 11 Probability and Statistics Homework 2 Theoretical Probability Answers. Theoretical probability is an important concept in probability and statistics. It refers to the likelihood or chance of an event occurring based on mathematical calculations and assumptions, rather than actual observations or experiments. ...
Properties of a binomial experiment (or Bernoulli trial) Homework; Section 5.1 introduced the concept of a probability distribution. The focus of the section was on discrete probability distributions (pdf). To find the pdf for a situation, you usually needed to actually conduct the experiment and collect data.
Definition 4.1.6 4.1. 6. Experimental Probabilities. P(A) = number of times A occurs number of times the experiment was repeated P ( A) = number of times A occurs number of times the experiment was repeated. For the event of getting a 6, the probability would by 163 1000 = 0.163 163 1000 = 0.163.