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3.2: Three Types of Probability

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• Page ID 36654

• Rachel Webb
• Portland State University

Learning Objectives

• Find theoretical probabilities
• Find empirical probabilities
• Find subjective probabilities

Probability is the likelihood of an event happening. Probabilities can be given as a percent, a decimal or a reduced fraction. The notation for the probability of event A is P( A ). Here are some important characteristics of probabilities:

• The probability of any event A is a number between 0 and 1:

0 ≤ P( A ) ≤ 1

• The sum of the probabilities of all of the outcomes in the sample space is 1:

P( A 1 ) + P( A 2 ) + … + P( A n ) = 1

• P( A ) = 0 means that event A will not happen
• P( A ) = 1 means that event A will definitely happen

There are three types of probability: theoretical, empirical, and subjective.

Classical Approach to Probability (Theoretical Probability)

Formula: theoretical probability.

\[P(A) = \dfrac{\text{Number of ways A can occur}}{\text{Number of different outcomes in S}}\]

The classical approach can only be used if each outcome has equal probability.

Example \(\PageIndex{1}\)

If an experiment consists of flipping a coin twice, compute the probability of getting exactly two heads.

There are 4 outcomes in the samples space, S = {HH, HT, TH, TT}. The event of getting exactly two heads is A = {HH}. The number of ways A can occur is 1. Thus P( A ) = \(\dfrac{1}{4}\).

Example \(\PageIndex{2}\)

If a random experiment consists of rolling a six-sided die, compute the probability of rolling a 4.

The sample space is S = {1, 2, 3, 4, 5, 6}. The event A is that you want is to get a 4, and the event space is A = {4}. Thus, in theory, the probability of rolling a 4 would be P( A ) = \(\dfrac{1}{6}\) = 0.1667.

Example \(\PageIndex{3}\)

Suppose you have an iPhone with the following songs on it: 5 Rolling Stones songs, 7 Beatles songs, 9 Bob Dylan songs, 4 Johnny Cash songs, 2 Carrie Underwood songs, 7 U2 songs, 4 Mariah Carey songs, 7 Bob Marley songs, 6 Bunny Wailer songs, 7 Elton John songs, 5 Led Zeppelin songs, and 4 Dave Matthews Band songs. The different genre that you have are rock from the ‘60s which includes Rolling Stones, Beatles, and Bob Dylan; country which includes Johnny Cash and Carrie Underwood; rock of the ‘90s includes U2 and Mariah Carey; reggae which includes Bob Marley and Bunny Wailer; rock of the ‘70s which includes Elton John and Led Zeppelin; and bluegrass/rock which includes Dave Matthews Band.

• What is the probability that you will hear a Bunny Wailer song?
• What is the probability that you will hear a song from the ‘60s?
• What is the probability that you will hear a reggae song?
• What is the probability that you will hear a song from the ‘90s or a bluegrass/rock song?
• What is the probability that you will hear an Elton John or a Carrie Underwood song?
• What is the probability that you will hear a country song or a U2 song?
• An iPhone in shuffle mode randomly picks the next song so you have no idea what the next song will be. Now you would like to calculate the probability that you will hear the type of music or the artist that you are interested in. The sample set is too difficult to write out, but you can figure it from looking at the number in each set and the total number. The total number of songs you have is 67. There are 4 Johnny Cash songs out of the 67 songs. Thus, P(Johnny Cash song) = \(\dfrac{4}{67}\) = 0.0597
• There are 6 Bunny Wailer songs. Thus, P(Bunny Wailer) = \(\dfrac{6}{67}\) = 0.0896.
• There are 5, 7, and 9 songs that are classified as rock from the ‘60s, which is a total of 21. Thus, P(rock from the ‘60s) = \(\dfrac{21}{67}\) = 0.3134.
• There are total of 13 songs that are classified as reggae. Thus, P(reggae) = \(\dfrac{13}{67}\) = 0.1940.
• There are 7 and 4 songs that are songs from the ‘90s and 4 songs that are bluegrass/rock, for a total of 15. Thus, P(rock from the ‘90s or bluegrass/rock) = \(\dfrac{15}{67}\) = 0.2239.
• There are 7 Elton John songs and 2 Carrie Underwood songs, for a total of 9. Thus, P(Elton John or Carrie Underwood song) =\(\dfrac{9}{67}\) = 0.1343.
• There are 6 country songs and 7 U2 songs, for a total of 13. Thus, P(country or U2 song) = \(\dfrac{13}{67}\) = 0.1940.

Empirical Probability (Experimental or Relative Frequency Probability)

Definition: empirical probability.

The experiment is performed many times and the number of times that event A occurs is recorded. Then the probability is approximated by finding the relative frequency.

\[P(A) = \dfrac{\text{Number of times A occurred}}{\text{Number of times the experiment was repeated}}\]

Example \(\PageIndex{4}\)

Suppose that the experiment is rolling a die. Find the empirical probability of rolling a 4.

The sample space is S = {1, 2, 3, 4, 5, 6}. The event A is that you want is to get a 4, and the event space is A = {4}. To do this, roll a die 10 times and count the number of times you roll a 4. When you do that, you get 4 two times. Based on this experiment, the probability of getting a 4 is 2 out of 10 or \(\dfrac{1}{5}\) = 0.2. To get more accuracy, repeat the experiment more times. It is easiest to put this information in a table, where n represents the number of times the experiment is repeated. When you put the number of 4s rolled divided by the number of times you repeat the experiment, you get the relative frequency. See the last column in Figure \(\PageIndex{1}\).

Figure \(\PageIndex{1}\): Trials for Die Experiment

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 4 is approximately 0.163. If you want more accuracy, then increase n even more by rolling the die more times.

These probabilities are called experimental probabilities since they are found by actually doing the experiment or simulation. They come about from the relative frequencies and give an approximation of the true probability.

The approximate probability of an event \(A\), notated as \(P(A)\), is

For the event of getting a 4, the probability would be P(4) = \(\dfrac{163}{1000}\) = 0.163

Definition: Law of Large Numbers

As n increases, the relative frequency tends toward the theoretical probability.

Figure \(\PageIndex{2}\) shows a graph of experimental probabilities as n gets larger and larger. 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 \(\PageIndex{2}\)

You can compute 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 of 5 children or the fairness of the die. This is due to the Law of Large Numbers, since the more times we repeat the experiment, the closer the experimental probabilities will get to the theoretical probabilities. For difficult theoretical probabilities, we can run computer simulations that can do an experiment repeatedly, many times, very quickly and come up with accurate estimates of the theoretical probability.

Example \(\PageIndex{5}\)

A fitness center coach kept track of members over the last year. They recorded if the person stretched before they exercised, and whether they sustained an injury. The following contingency table shows their results. Select one member at random and find the following probabilities.

• Find the probability that a member sustained an injury.
• Find the probability that a member did not stretch.
• Find the probability that a member sustained an injury and did not stretch.
• Find the totals for each row, column, and grand total.

Next, find the relative frequencies by dividing each number by the total of 400.

Using the formula for probability. we get P(Injury) = \(\dfrac{\text{Number of injuries}}{\text{Total number of people}}\) = \(\dfrac{73}{400}\) = 0.1825.

Using the table, we can get the same answer very quickly by just taking the column total under Injury to get 0.1825. As we get more complicated probability questions, these contingency tables will help organize your data.

• Using the relative frequency contingency table, take the total of the row for all the members that did not stretch and we get the P(Did Not Stretch) = 0.195.
• Using the relative frequency contingency table, take the intersection of the injury column with the did not stretch row and we get P(Injury and Did Not Stretch) = 0.0525.

Subjective Probability

Definition: subjective probability.

Subjective probability is the probability of event A estimated using previous knowledge and is someone’s opinion.

Example \(\PageIndex{6}\)

Find the probability of meeting Dolly Parton.

I estimate the probability of meeting Dolly Parton to be \(1.2 \times 10^{-9}\) = 1.2 E-9 \(\approx\) 0.0000000012. This is a very small probability and essentially means that the probability is 0 and meeting Dolly Parton will not happen.

Example \(\PageIndex{7}\)

What is the probability it will rain tomorrow?

A weather reporter looks at several forecasts, uses their expert knowledge of the region, and reports the probability that it will rain tomorrow is 80%.

Theoretical Probability & Experimental Probability

Related Pages Probability Tree Diagrams Probability Without Replacement Probability Word Problems More Lessons On Probability

In these lessons, we will look into experimental probability and theoretical probability.

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).

How to find and use experimental probability?

The following video gives another example of experimental probability.

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.

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

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.

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

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?

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?

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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

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3.4: Introduction to Probability

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• Page ID 146478

• Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia
• Rio Hondo College

4.1 Learning Objectives

• Describe theoretical, empirical, and subjective probability
• Distinguish among the three uses of probability

The probability of a specified event is the chance or likelihood that it will occur. There are several ways of viewing probability.

One would be experimental in nature, where we repeatedly conduct an experiment. Suppose we flipped a coin over and over and over again and it came up heads about half of the time; we would expect that in the future whenever we flipped the coin it would turn up heads about half of the time. When a weather reporter says “there is a 10% chance of rain tomorrow,” she is basing that on prior evidence; that out of all days with similar weather patterns, it has rained on 1 out of 10 of those days.

Conduct an experiment to determine the probability of the spinner landing on 1?

Using the experimental method, suppose out of 10,000 spins, 1,711 of those landed on 1. This is normally done on a computer application that can randomly generate outcomes of each spin to avoid someone having to spin the spinner 10,000 times.

To calculate this probability, divide the number of times 1 has occurred which is 1,711 by the number of times the spinner was spun which was 10,000.

The result is 0.1711 or approximately 17% of the time. So 17 times out of 100 spins one would expect the spinner to land on the number 1.

Another view would be subjective in nature, in other words an educated guess or opinion. But this is just a guess, with no way to verify its accuracy, and depending upon how educated the educated guesser is, a subjective probability may not be worth very much.

Determine the probability that the Seattle Mariners would win their next baseball game.

It would be impossible to conduct an experiment where the same two teams played each other repeatedly, each time with the same starting lineup and starting pitchers, each starting at the same time of day on the same field under the precisely the same conditions.

Since there are so many variables to take into account, someone familiar with baseball and with the two teams involved might make an educated guess that there is a 75% chance the Mariners will win the game; that is, if   the same two teams were to play each other repeatedly under identical conditions, the Mariners would win about three out of every four games.

Definition: Probabilities

Empirical Probability uses the results of an experiment to predict the percent chance an event could occur.

Subjective Probability uses intuition or guesswork to predict the percent chance an event could occur.

Theoretical Probability uses the number of possible desired outcomes of an event compared to the number of all possible outcomes of an event to predict the percent chance an event could occur.

Theoretical Probability is defined mathematically as follows:

Suppose there is a situation with \(n\) equally likely possible outcomes and that m of those \(n\) outcomes correspond to a particular event; then the probability of that event is defined as \(\frac{m}{n}\).

We will return to the empirical and subjective probabilities from time to time, but in this course we will mostly be concerned with theoretical probability.

Theoretical and Experimental Probability

More specific topics in theoretical and experimental probability.

• Simple and Theoretical Probability
• Experimental and Geometric Probability

Popular Tutorials in Theoretical and Experimental 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!

How Do You Use a Simulation to Solve a Problem?

Simulators are a great way to model an experiment without actually performing the experiment in real life. This tutorial looks at using a simulator to figure out what might happen if you randomly guessed on a true/false quiz.

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 Experimental Probability?

Do real life situations always work out the way your mathematical models tell you they should? No! This tutorial describes how experimental probability differs from theoretical 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 probability and statistics :.

• Permutations and Combinations
• Probabilities of Compound Events
• Measures of Central Tendency and Dispersion
• Populations and Sampling
• Statistics and Hypothesis Testing
• The Normal Distribution
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3. When would you expect the experimental probability of an event to be closest to its thecretical probability? A. The results of 10 trials are used to calculate the theoretical probability B. The results of 10 trials are used to calculate the expermental probability C. The results of 100 trials are used to calculate the experimental propability. D. None the statements are true, because the experimental probability is unrelated to the theoretical probability.

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