examples of experimental probability in math

Experimental Probability

The chance or occurrence of a particular event is termed its probability. The value of a probability lies between 0 and 1 which means if it is an impossible event, the probability is 0 and if it is a certain event, the probability is 1. The probability that is determined on the basis of the results of an experiment is known as experimental probability. This is also known as empirical probability.

What is Experimental Probability?

Experimental probability is a probability that is determined on the basis of a series of experiments. A random experiment is done and is repeated many times to determine their likelihood and each repetition is known as a trial. The experiment is conducted to find the chance of an event to occur or not to occur. It can be tossing a coin, rolling a die, or rotating a spinner. In mathematical terms, the probability of an event is equal to the number of times an event occurred ÷ the total number of trials. For instance, you flip a coin 30 times and record whether you get a head or a tail. The experimental probability of obtaining a head is calculated as a fraction of the number of recorded heads and the total number of tosses. P(head) = Number of heads recorded ÷ 30 tosses.

Experimental Probability Formula

The experimental probability of an event is based on the number of times the event has occurred during the experiment and the total number of times the experiment was conducted. Each possible outcome is uncertain and the set of all the possible outcomes is called the sample space. The formula to calculate the experimental probability is: P(E) = Number of times an event occurs/Total number of times the experiment is conducted

Consider an experiment of rotating a spinner 50 times. The table given below shows the results of the experiment conducted. Let us find the experimental probability of spinning the color - blue.

The experimental probability of spinning the color blue = 10/50 = 1/5 = 0.2 = 20%

Experimental Probability vs Theoretical Probability

Experimental results are unpredictable and may not necessarily match the theoretical results. The results of experimental probability are close to theoretical only if the number of trials is more in number. Let us see the difference between experimental probability and theoretical probability.

Experimental Probability Examples

Here are a few examples from real-life scenarios.

a) The number of cookies made by Patrick per day in this week is given as 4, 7, 6, 9, 5, 9, 5.

Based on this data, what is the reasonable estimate of the probability that Patrick makes less than 6 cookies the next day?

P(< 6 cookies) = 3/7 = 0.428 = 42%

b) Find the reasonable estimate of the probability that while ordering a pizza, the next order will not be of a pepperoni topping.

Based on this data , the reasonable estimate of the probability that the next type of toppings that would get ordered is not a pepperoni will be 15/20 = 3/4 = 75%

Related Sections

• Card Probability
• Conditional Probability Calculator
• Binomial Probability Calculator
• Probability Rules
• Probability and Statistics

Important Notes

• The sum of the experimental probabilities of all the outcomes is 1.
• The probability of an event lies between 0 and 1, where 0 is an impossible event and 1 denotes a certain event.
• Probability can also be expressed in percentage.

Examples on Experimental Probability

Example 1: The following table shows the recording of the outcomes on throwing a 6-sided die 100 times.

Find the experimental probability of: a) Rolling a four; b) Rolling a number less than four; c) Rolling a 2 or 5

Experimental probability is calculated by the formula: Number of times an event occurs/Total number of trials

a) Rolling a 4: 17/100 = 0.17

b) Rolling a number less than 4: 56/100 = 0.56

c) Rolling a 2 or 5: 31/100 = 0.31

Example 2: The following set of data shows the number of messages that Mike received recently from 6 of his friends. 4, 3, 2, 1, 6, 8. Based on this, find the probability that Mike will receive less than 2 messages next time.

Mike has received less than 2 messages from 2 of his friends out of 6.

Therefore, P(<2) = 2/6 = 1/3

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Practice Questions on Experimental Probability

Frequently asked questions (faqs), how do you find the experimental probability.

The experimental probability of an event is based on actual experiments and the recordings of the events. It is equal to the number of times an event occurred divided by the total number of trials.

What is the Experimental Probability of rolling a 6?

The experimental probability of rolling a 6 is 1/6. A die has 6 faces numbered from 1 to 6. Rolling the die to get any number from 1 to 6 is the same and the probability (of getting a 6) = Number of favorable outcomes/ total possible outcomes = 1/6.

What is the Difference Between Theoretical and Experimental Probability?

Theoretical probability is what is expected to happen and experimental probability is what has actually happened in the experiment.

Do You Simplify Experimental Probability?

Yes, after finding the ratio of the number of times the event occurred to the total number of trials conducted, the fraction which is obtained is simplified.

Which Probability is More Accurate, Theoretical Probability or Experimental Probability?

Theoretical probability is more accurate than experimental probability. The results of experimental probability are close to theoretical only if the number of trials are more in number.

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What is experimental probability? 

Practice questions, experimental probability – explanation & examples.

Experimental probability is the probability determined based on the results from performing the particular experiment. 

In this lesson we will go through:

• The meaning of experimental probability
• How to find experimental probability

The ratio of the number of outcomes favorable to an event to the total number of trials of the experiment.

Experimental Probability can be expressed mathematically as: 

$P(\text{E}) = \frac{\text{number of outcomes favorable to an event}}{\text{total number of trials of the experiment}}$

Let’s go back to the die tossing example. If after 12 throws you get one 6, then the experimental probability is $\frac{1}{12}$.  You can compare that to the theoretical probability. The theoretical probability of getting a 6 is $\frac{1}{6}$. This means that in 12 throws we would have expected to get 6 twice. 

Similarly, if in those 12 tosses you got a 1 five times, the experimental probability is $\frac{5}{12}$. 

How do we find experimental probability?

Now that we understand what is meant by experimental probability, let’s go through how it is found. 

To find the experimental probability of an event, divide the number of observed outcomes favorable to the event by the total number of trials of the experiment. 

Let’s go through some examples. 

Example 1:  There are 20 students in a class. Each student simultaneously flipped one coin. 12 students got a Head. From this experiment, what was the experimental probability of getting a head?

Number of coins showing Heads: 12

Total number of coins flipped: 20

$P(\text{Head}) = \frac{12}{20} = \frac{3}{5}$ 

Example 2:  The tally chart below shows the number of times a number was shown on the face of a tossed die. 

a. What was the probability of a 3 in this experiment?

b. What was the probability of a prime number?

First, sum the numbers in the frequency column to see that the experiment was performed 30 times. Then find the probabilities of the specified events. 

a. Number of times 3 showed = 7

Number of tosses = 30

$P(\text{3}) = \frac{7}{30}$ 

b. Frequency of primes = 6 + 7 + 2 = 15

Number of trials = 30 

$P(\text{prime}) = \frac{15}{30} = \frac{1}{2}$

Experimental probability can be used to predict the outcomes of experiments. This is shown in the following examples. 

Example 3: The table shows the attendance schedule of an employee for the month of May.

a. What is the probability that the employee is absent? 

b. How many times would we expect the employee to be present in June?

a. The employee was absent three times and the number of days in this experiment was 31. Therefore:

$P(\text{Absent}) = \frac{3}{31}$

b.  We expect the employee to be absent

$\frac{3}{31} × 30 = 2.9 ≈ 3$ days in June 

Example 4:  Tommy observed the color of cars owned by people in his small hometown. Of the 500 cars in town, 10 were custom colors, 100 were white, 50 were red, 120 were black, 100 were silver, 60 were blue, and 60 were grey. 

a. What is the probability that a car is red?

b. If a new car is bought by someone in town, what color do you think it would be? Explain. 

a. Number of red cars = 50 

Total number of cars = 500 

$P(\text{red car}) = \frac{50}{500} = \frac{1}{10}$ 

b. Based on the information provided, it is most likely that the new car will be black. This is because it has the highest frequency and the highest experimental probability. 

Now it is time for you to try these examples. 

The table below shows the colors of jeans in a clothing store and their respective frequencies. Use the table to answer the questions that follow.

• What is the probability of selecting a brown jeans?
• What is the probability of selecting a blue or a white jeans?

On a given day, a fast food restaurant notices that it sold 110 beef burgers, 60 chicken sandwiches, and 30 turkey sandwiches. From this observation, what is the experimental probability that a customer buys a beef burger?

Over a span of 20 seasons, a talent competition notices the following. Singers won 12 seasons, dancers won 2 seasons, comedians won 3 seasons, a poet won 1 season, and daring acts won the other 2 seasons. 

a. What is the experimental probability of a comedian winning  a season?

b. From the next 10 seasons, how many winners do you expect to be dancers?

Try this at home! Flip a coin 10 times. Record the number of tails you get. What is your P(tail)?

Number of brown jeans = 25

Total Number of jeans = 125

$P(\text{brown}) = \frac{25}{125}  = \frac{1}{5}$

Number of jeans that are blue or white = 75 + 20 = 95

$P(\text{blue or white}) = \frac{95}{125} = \frac{19}{25}$

Number of beef burgers = 110 

Number of burgers (or sandwiches) sold = 200 

$P(\text{beef burger}) = \frac{110}{200} = \frac{11}{20}$ 

a. Number of comedian winners = 3

Number of seasons = 20 

$P(\text{comedian}) = \frac{3}{20}$ 

b. First find the experimental probability that the winner is a dancer. 

Number of winners that are dancers = 2 

$P(\text{dancer}) = \frac{2}{20} = \frac{1}{10}$ 

Therefore we expect 

$\frac{1}{10} × 10 = 1$ winner to be a dancer in the next 10 seasons.

To find your P(tail) in 10 trials, complete the following with the number of tails you got. 

$P(\text{tail}) = \frac{\text{number of tails}}{10}$ 

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Experimental Probability

Experimental probability: introduction, experimental probability: definition, experimental probability formula, solved examples, practice problems, frequently asked questions.

In mathematics, probability refers to the chance of occurrence of a specific event. Probability can be measured on a scale from 0 to 1. The probability is 0 for an impossible event. The probability is 1 if the occurrence of the event is certain.

There are two approaches to study probability: experimental and theoretical. 

Suppose you and your friend toss a coin to decide who gets the first turn to ride a new bicycle. You choose “heads” and your friend chooses “tails.” 

Can you guess who will win? No! You have $\frac{1}{2}$ a chance of winning and so does your friend. This is theoretical since you are predicting the outcome based on what is expected to happen and not on the basis of outcomes of an experiment.

So, what is the experimental probability? Experimental probability is calculated by repeating an experiment and observing the outcomes. Let’s understand this a little better.

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Experimental probability, or empirical probability, is the probability calculated by performing actual experiments and gathering or recording the necessary information. How would you define an experiment? The math definition of an experiment is “a process or procedure that can be repeated and that has a set of well-defined possible results or outcomes.”

Consider the same example. Suppose you flip the coin 50 times to see whether you get heads or tails, and you record the outcomes. Suppose you get heads 20 times and tails 30 times. Then the probability calculated using these outcomes is experimental probability. Here, t he experimental meaning is connected with such experiments used to determine the probability of an event.

Now that you know the meaning of experimental probability, let’s understand its formula.

Experimental Probability for an Event A can be calculated as follows:

P(E) $= \frac{Number of occurance of the event A}{Total number of trials}$

Let’s understand this with the help of the last example. 

A coin is flipped a total of 50 times. Heads appeared 20 times. Now, what is the experimental probability of getting heads?

E xperimental probability of getting heads $= \frac{Number of occurrences}{Total number of trials}$

P (Heads) $= \frac{20}{50} = \frac{2}{5}$

P (Tails) $= \frac{30}{50} = \frac{3}{5}$

Experimental Probability vs. Theoretical Probability

Theoretical probability expresses what is expected. On the other hand, experimental probability explains how frequently an event occurred in an experiment.

If you roll a die, the theoretical probability of getting any particular number, say 3, is $\frac{1}{6}$. 

However, if you roll the die 100 times and record how many times 3 appears on top, say 65 times, then the experimental probability of getting 3 is $\frac{65}{100}$.

Theoretical probability for Event A can be calculated as follows:

P(A) $= \frac{Number of outcomes favorable to Event A}{Number of possible outcomes}$

In the example of flipping a coin, the theoretical probability of the occurrence of heads (or tails) on tossing a coin is

P(H) $= \frac{1}{2}$ and  P(T) $= \frac{1}{2}$ (since possible outcomes are $2 -$ head or tail)

Experimental Probability: Examples

Let’s take a look at some of the examples of experimental probability .

Example 1: Ben tried to toss a ping-pong ball in a cup using 10 trials, out of which he succeeded 4 times. 

P(win) $= \frac{Number of success}{Number of trials}$

             $= \frac{4}{10}$

             $= \frac{2}{5}$

Example 2: Two students are playing a game of die. They want to know how many times they land on 2 on the dice if the die is rolled 20 times in a row. 

The experimental probability of rolling a 2 

$= \frac{Number of times 2 appeared}{Number of trials}$

$= \frac{5}{20}$

$= \frac{1}{4}$

1. Probability of an event always lies between 0 and 1.

2. You can also express the probability as a decimal and a percentage.

Experimental probability is a probability that is determined by the results of a series of experiments. Learn more such interesting concepts at SplashLearn .

1. Leo tosses a coin 25 times and observes that the “head” appears 10 times. What is the experimental probability of getting a head?

 P(Head) $= \frac{Number of times heads appeared}{Total number of trials}$

               $= \frac{10}{25}$

               $= \frac{2}{5}$

               $= 0.4$

2. The number of cakes a baker makes per day in a week is given as 7, 8, 6, 10, 2, 8, 3. What is the probability that the baker makes less than 6 cakes the next day?

Solution: 

Number of cakes baked each day in a week $= 7, 8, 6, 10, 2, 8, 3$

Out of 7 days, there were 2 days (highlighted in bold) on which the baker made less than 6 cookies.

P$(< 6 $cookies$) = \frac{2}{7}$

3. The chart below shows the number of times a number was shown on the face of a tossed die. What was the probability of getting a 3 in this experiment?

Number of times 3 showed $= 7$

Number of tosses $= 30$

P(3) $= \frac{7}{30}$

4. John kicked a ball 20 times. He kicked 16 field goals and missed 4 times . What is the experimental probability that John will kick a field goal during the game?

Solution:  

John succeeded in kicking 16 field goals. He attempted to kick a field goal 20 times. 

So, the number of trials $= 20$

John’s experimental probability of kicking a field goal $= \frac{Successful outcomes} {Trials attempted} = \frac{16}{20}$ 

$= \frac{4}{5}$

$= 0.8$ or $80%$

5. James recorded the color of bikes crossing his street. Of the 500 bikes, 10 were custom colors, 100 were white, 50 were red, 120 were black, 100 were silver, 60 were blue, and 60 were gray. What is the probability that the car crossing his street is white?

Number of white bikes $= 100$ 

Total number of bikes $= 500$

P(white bike) $=  \frac{100}{500} = \frac{1}{5}$

Attend this quiz & Test your knowledge.

In a class, a student is chosen randomly in five trials to participate in 5 different events. Out of chosen students, 3 were girls and 2 were boys. What is the experimental probability of choosing a boy in the next event?

A manufacturer makes 1000 tablets every month. after inspecting 100 tablets, the manufacturer found that 30 tablets were defective. what is the probability that you will buy a defective tablet, the 3 coins are tossed 1000 times simultaneously and we get three tails $= 160$, two tails $= 260$, one tail $= 320$, no tails $= 260$. what is the probability of occurrence of two tails, the table below shows the colors of shirts sold in a clothing store on a particular day and their respective frequencies. use the table to answer the questions that follow. what is the probability of selling a blue shirt.

Jason leaves for work at the same time each day. Over a period of 327 working days, on his way to work, he had to wait for a train at the railway crossing for 68 days. What is the experimental probability that Jason has to wait for a train on his way to work?

What is the importance of experimental probability?

Experimental probability is widely used in research and experiments in various fields, such as medicine, social sciences, investing, and weather forecasting.

Is experimental probability always accurate?

Predictions based on experimental probability are less reliable than those based on theoretical probability.

Can experimental probability change every time the experiment is performed?

Since the experimental probability is based on the actual results of an experiment, it can change when the results of an experiment change.

What is theoretical probability?

The theoretical probability is calculated by finding the ratio of the number of favorable outcomes to the total number of probable outcomes.

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Experimental Probability

Here we will learn about experimental probability, including using the relative frequency and finding the probability distribution.

There are also probability distribution worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is experimental probability?

Experimental probability i s the probability of an event happening based on an experiment or observation.

To calculate the experimental probability of an event, we calculate the relative frequency of the event.

We can also express this as R=\frac{f}{n} where R is the relative frequency, f is the frequency of the event occurring, and n is the number of trials of the experiment.

If we find the relative frequency for all possible events from the experiment we can write the probability distribution for that experiment.

The relative frequency, experimental probability and empirical probability are the same thing and are calculated using the data from random experiments. They also have a key use in real-life problem solving.

For example, Jo made a four-sided spinner out of cardboard and a pencil.

She spun the spinner 50 times. The table shows the number of times the spinner landed on each of the numbers 1 to 4. The final column shows the relative frequency.

The relative frequencies of all possible events will add up to 1.

This is because the events are mutually exclusive.

Step-by-step guide: Mutually exclusive events

Experimental probability vs theoretical probability

You can see that the relative frequencies are not equal to the theoretical probabilities we would expect if the spinner was fair.

If the spinner is fair, the more times an experiment is done the closer the relative frequencies should be to the theoretical probabilities.

In this case the theoretical probability of each section of the spinner would be 0.25, or \frac{1}{4}.

Step-by-step guide: Theoretical probability

How to find an experimental probability distribution

In order to calculate an experimental probability distribution:

Draw a table showing the frequency of each outcome in the experiment.

Determine the total number of trials.

Write the experimental probability (relative frequency) of the required outcome(s).

Explain how to find an experimental probability distribution

Experimental probability worksheet

Get your free experimental probability worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Related lessons on   probability distribution

Experimental probability  is part of our series of lessons to support revision on  probability distribution . You may find it helpful to start with the main probability distribution lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Probability distribution
• Relative frequency
• Expected frequency

Experimental probability examples

Example 1: finding an experimental probability distribution.

A 3 sided spinner numbered 1,2, and 3 is spun and the results recorded.

Find the probability distribution for the 3 sided spinner from these experimental results.

A table of results has already been provided. We can add an extra column for the relative frequencies.

2 Determine the total number of trials

3 Write the experimental probability (relative frequency) of the required outcome(s).

Divide each frequency by 110 to find the relative frequencies.

Example 2: finding an experimental probability distribution

A normal 6 sided die is rolled 50 times. A tally chart was used to record the results.

Determine the probability distribution for the 6 sided die. Give your answers as decimals.

Use the tally chart to find the frequencies and add a row for the relative frequencies.

The question stated that the experiment had 50 trials. We can also check that the frequencies add to 50.

Divide each frequency by 50 to find the relative frequencies.

Example 3: using an experimental probability distribution

A student made a biased die and wanted to find its probability distribution for use in a game. They rolled the die 100 times and recorded the results.

By calculating the probability distribution for the die, determine the probability of the die landing on a 3 or a 4.

The die was rolled 100 times.

We can find the probability of rolling a 3 or a 4 by adding the relative frequencies for those numbers.

P(3 or 4) = 0.22 + 0.25 = 0.47

Example 4: calculating the relative frequency without a known frequency of outcomes

A research study asked 1200 people how they commute to work. 640 travelled by car, 174 used the bus, and the rest walked. Determine the relative frequency of someone not commuting to work by car.

Writing the known information into a table, we have

We currently do not know the frequency of people who walked to work. We can calculate this as we know the total frequency.

The number of people who walked to work is equal to

1200-(640+174)=386.

We now have the full table,

The total frequency is 1200.

Divide each frequency by the total number of people (1200), we have

The relative frequency of someone walking to work is 0.321\dot{6} .

How to find a frequency using an experimental probability

In order to calculate a frequency using an experimental probability:

Multiply the total frequency by the experimental probability.

Explain how to find a frequency using an experimental probability

Example 5: calculating a frequency

A dice was rolled 300 times. The experimental probability of rolling an even number is \frac{27}{50}. How many times was an even number rolled?

An even number was rolled 162 times.

Example 6: calculating a frequency

A bag contains different coloured counters. A counter is selected at random and replaced back into the bag 240 times. The probability distribution of the experiment is given below.

Determine the number of times a blue counter was selected.

As the events are mutually exclusive, the sum of the probabilities must be equal to 1. This means that we can determine the value of x.

1-(0.4+0.25+0.15)=0.2

The experimental probability (relative frequency) of a blue counter is 0.2.

Multiplying the total frequency by 0.1, we have

240 \times 0.2=48.

A blue counter was selected 48 times.

Common misconceptions

• Forgetting the differences between theoretical and experimental probability

It is common to forget to use the relative frequencies from experiments for probability questions and use the theoretical probabilities instead. For example, they may be asked to find the probability of a die landing on an even number based on an experiment and the student will incorrectly answer it as 0.5.

• The relative frequency is not an integer

The relative frequency is the same as the experimental probability. This value is written as a fraction, decimal or percentage, not an integer.

Practice experimental probability questions

1. A coin is flipped 80 times and the results recorded.

Determine the probability distribution of the coin.

As the number of tosses is 80, dividing the frequencies for the number of heads and the number of tails by 80, we have

2. A 6 sided die is rolled 160 times and the results recorded.

Determine the probability distribution of the die. Write your answers as fractions in their simplest form.

Dividing the frequencies of each number by 160, we get

3. A 3 -sided spinner is spun and the results recorded.

Find the probability distribution of the spinner, giving you answers as decimals to 2 decimal places.

Dividing the frequencies of each colour by 128 and simplifying, we have

4. A 3 -sided spinner is spun and the results recorded.

Find the probability of the spinner not landing on red. Give your answer as a fraction.

Add the frequencies of blue and green and divide by 128.

5. A card is picked at random from a deck and then replaced. This was repeated 4000 times. The probability distribution of the experiment is given below.

How many times was a club picked?

6. Find the missing frequency from the probability distribution.

The total frequency is calculated by dividing the frequency by the relative frequency.

Experimental probability GCSE questions

1. A 4 sided spinner was spun in an experiment and the results recorded.

(a) Complete the relative frequency column. Give your answers as decimals.

(b) Find the probability of the spinner landing on a square number.

Total frequency of 80.

2 relative frequencies correct.

All 4 relative frequencies correct 0.225, \ 0.2, \ 0.3375, \ 0.2375.

Relative frequencies of 1 and 4 used.

0.4625 or equivalent

2. A 3 sided spinner was spun and the results recorded.

Complete the table.

Process to find total frequency or use of ratio with 36 and 0.3.

3. Ben flipped a coin 20 times and recorded the results.

(a) Ben says, “the coin must be biased because I got a lot more heads than tails”.

Comment on Ben’s statement.

(b) Fred takes the same coin and flips it another 80 times and records the results.

Use the information to find a probability distribution for the coin.

Stating that Ben’s statement may be false.

Mentioning that 20 times is not enough trials.

Evidence of use of both sets of results from Ben and Fred.

Process of dividing by 100.

P(heads) = 0.48 or equivalent

P(tails) = 0.52 or equivalent

Learning checklist

You have now learned how to:

• Use a probability model to predict the outcomes of future experiments; understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size

The next lessons are

• How to calculate probability
• Combined events probability
• Describing probability

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Experimental probability

Experimental probability (EP), also called empirical probability or relative frequency , is probability based on data collected from repeated trials.

Experimental probability formula

Let n represent the total number of trials or the number of times an experiment is done. Let p represent the number of times an event occurred while performing this experiment n times.

Example #1: A manufacturer makes 50,000 cell phones every month. After inspecting 1000 phones, the manufacturer found that 20 phones are defective. What is the probability that you will buy a phone that is defective? Predict how many phones will be defective next month.

The total number of times the experiment is conducted is n = 1000

The number of times an event occurred is p  = 20 

Experimental probability is performed when authorities want to know how the public feels about a matter. Since it is not possible to ask every single person in the country, they may conduct a survey by asking a sample of the entire population. This is called population sampling. Example #2 is an example of this situation.

There are about 319 million people living in the USA. Pretend that a survey of 1 million people revealed that 300,000 people think that all cars should be electric. What is the probability that someone chosen randomly does not like electric car? How many people like electric cars?

Notice that the number of people who do not like electric cars is 1000000 - 300000 = 700000

Difference between experimental probability and theoretical probability

You can argue the same thing using a die, a coin, and a spinner. We will though use a coin and a spinner to help you see the difference.

Using a coin 

In theoretical probability, we say that "each outcome is equally likely " without the actual experiment. For instance, without  flipping a coin, you know that the outcome could either be heads or tails.  If the coin is not altered, we argue that each outcome (heads or tails) is equally likely. In other words, we are saying that in theory or (supposition, conjecture, speculation, assumption, educated guess) the probability to get heads is 50% or the probability to get tails in 50%. Since you did not actually flip the coin, you are making an assumption based on logic.

The logic is that there are 2 possible outcomes and since you are choosing 1 of the 2 outcomes, the probability is 1/2 or 50%. This is theoretical probability or guessing probability or probability based on assumption.

In the example above about flipping a coin, suppose you are looking for the probability to get a head. 

Then, the number of favorable outcomes is 1 and the number of possible outcomes is 2.

In experimental probability,  we want to take the guess work out of the picture, by doing the experiment to see how many times heads or teals will come up. If you flip a coin 1000 times, you might realize that it landed on heads only 400 times. In this case, the probability to get heads is only 40%. 

Your experiment may not even show tails until after the 4th flip and yet in the end you ended up with more tails than heads. 

If you repeat the experiment another day, you may find a completely different result. May be this time the number of heads is 600 and the number of tails is 400.

Using a spinner

Suppose a spinner has four equal-sized sections that are red, green, black, and yellow. 

In theoretical probability, you will not spin the spinner. Instead, you will say that the probability to get green is one-fourth or 25%. Why 25%? The total number of outcomes is 4 and the number of favorable outcomes is 1.

1/4 = 0.25 = 25%

However, in experimental probability, you may decide to spin the spinner 50 times or even more to see how many times you will get each color.

Suppose you spin the spinner 50 times. It is quite possible that you may end up with the result shown below:

Red: 10 Green: 15 Black: 5 Yellow: 20

Now, the probability to get green is 15/50 = 0.3 = 30%

As you can see, experimental probability is based more on facts, data collected, experiment or research!

Theoretical probability

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Experimental Probability – Formula, Definition With Examples

Updated on January 12, 2024

Welcome to another exciting journey with us at Brighterly , where we make the learning of complex mathematical concepts a fun and engaging process. Today, we embark on a venture into the world of experimental probability, a vital aspect of mathematics that breathes life into numbers through practical, real-world experiences. But what exactly is experimental probability, and how does it differ from theoretical probability? How can we calculate experimental probability, and how is it applicable in our everyday lives? This article aims to answer these questions and more, unraveling the mysteries of experimental probability in an easy-to-understand and approachable manner.

Our trip into the world of experimental probability will cover the core concepts, definitions, and the all-important formula that underpins this fascinating area of mathematics. We’ll take a step back to appreciate the broader context of probability before focusing our lens on experimental probability, understanding its properties and contrasts with theoretical probability. With plenty of examples and practice problems, you’ll have a firm grasp on experimental probability, ready to see and use it in the world around you!

What Is Experimental Probability?

Probability, as a field of mathematics, often focuses on predicting the likelihood of certain events. However, it’s important to note that there are two main types of probability: theoretical and experimental. In this article, we will zero in on experimental probability.

Experimental probability, also known as empirical probability, is all about actual experiments and real-world observations. The main idea behind experimental probability is that it calculates the chances of an event happening based on the actual results of an experiment. This method of calculation is particularly interesting because it revolves around practical events that have already occurred, rather than theoretical or hypothetical situations.

In experimental probability, we conduct a certain experiment multiple times and observe the number of times a specific event occurs. This might sound quite complex, but we’ll dive into this concept with a greater depth in the upcoming sections, making it easily understandable.

Definition of Probability

Before we delve into experimental probability, let’s take a step back and understand the basic concept of probability. Probability is defined as a branch of mathematics that measures the likelihood of events to occur. It’s expressed as a number between 0 and 1, where 0 indicates impossibility and 1 signifies certainty.

For example, consider flipping a fair coin. The probability of landing a “heads” is 1 out of 2, or 0.5, meaning there’s a 50% chance to get a “heads”. The same applies to “tails”.

Understanding probability can help us make predictions about the outcomes of a random event and aids in making informed decisions in various aspects of life including gaming, statistics, and even weather forecasting.

Definition of Experimental Probability

Moving on to experimental probability, it is defined as the ratio of the number of times an event occurs to the total number of trials or times the activity is performed. The experimental probability of an event is calculated by conducting an experiment and recording the results.

For instance, let’s say we roll a dice 100 times, and the number “4” comes up 15 times. Here, the experimental probability of rolling a “4” would be the number of successful outcomes (rolling a “4”) divided by the total number of outcomes (total dice rolls), or 15/100 = 0.15.

In other words, experimental probability is the actual probability obtained from the direct observation or testing during an experiment. Unlike theoretical probability, it doesn’t rely on the inherent nature of the experiment, but rather on the actual data collected.

Properties of Probability

Understanding the properties of probability can provide us with insights about how probability functions. Here are some of the essential properties:

• The probability of an event ranges from 0 to 1.
• The sum of probabilities of all possible outcomes is always 1.
• The probability of the complement of an event (an event not happening) is 1 minus the probability of the event.
• If two events are mutually exclusive (they can’t occur at the same time), the probability of either event occurring is the sum of their individual probabilities.

These properties provide a foundational understanding of how probability works, whether it’s theoretical or experimental probability.

Properties of Experimental Probability

The properties of experimental probability are closely tied to those of theoretical probability, but with an emphasis on the data collected through experimentation. Here are the primary properties:

• Experimental probability also ranges from 0 to 1. An experimental probability of 0 means the event never happened in the experiment, and a probability of 1 means the event always occurred.
• As more trials are conducted, the experimental probability tends to approach the theoretical probability, given that the experiment is unbiased. This is known as the law of large numbers.
• Like in theoretical probability, the sum of experimental probabilities of all possible outcomes is 1.

Understanding these properties can greatly aid in interpreting the results of experiments and the likelihood of outcomes.

Difference Between Theoretical and Experimental Probability

The primary difference between theoretical and experimental probability lies in their calculation and interpretation. Theoretical probability is based on the possible outcomes in theory. It assumes that all outcomes are equally likely, which isn’t always the case in real-world scenarios.

On the other hand, experimental probability is based on actual data collected from performed experiments. It deals with the frequency of occurrence of an event, providing a more empirical perspective on probability. For example, in theory, the probability of rolling a “6” on a fair die is 1/6. However, in an actual experiment of, say, 60 rolls, we might roll a “6” only 8 times. The experimental probability then becomes 8/60 or 0.1333.

Formula of Experimental Probability

The formula of experimental probability is quite straightforward:

By using this formula, we can calculate the experimental probability of an event based on the results of an actual experiment or observation.

Understanding the Formula of Experimental Probability

To understand the formula of experimental probability, let’s revisit the dice rolling example. If you roll a die 100 times, and the number “4” comes up 20 times, then the experimental probability of rolling a “4” is:

Experimental Probability = Number of times event occurs / Total number of trials

Experimental Probability = 20 / 100 = 0.2

Hence, based on the results of this experiment, the experimental probability of rolling a “4” is 0.2 or 20%.

This formula essentially calculates the frequency of occurrence of an event in an experiment, providing a realistic interpretation of probability.

Calculating Experimental Probability Using the Formula

Let’s consider another example to illustrate the calculation of experimental probability using the formula. Imagine you’re shooting basketball hoops. You take 30 shots and make 18 of them. What’s the experimental probability of making a shot?

Applying the formula, we get:

Experimental Probability = 18 / 30 = 0.6

So, the experimental probability of making a shot, based on this experiment, is 0.6 or 60%.

Practice Problems on Experimental Probability

To better understand how to calculate experimental probability, let’s work through some practice problems:

• A spinner with 8 equal sections numbered 1 to 8 is spun 50 times. The number 3 comes up 7 times. What is the experimental probability of the spinner landing on 3?
• In a school, a survey of what pet each student has at home is conducted. Out of 200 students, 45 have dogs. What is the experimental probability that a randomly selected student has a dog?
• In a bag of 100 marbles, 25 are red, and the rest are blue. If you randomly select a marble, replace it, and repeat this 100 times, and you get a red marble 28 times, what is the experimental probability of drawing a red marble?
• Experimental Probability = 7 / 50 = 0.14
• Experimental Probability = 45 / 200 = 0.225
• Experimental Probability = 28 / 100 = 0.28

And that wraps up our enlightening exploration of experimental probability! With Brighterly, we’ve unpacked this fascinating mathematical concept, revealing its significance and wide-ranging applications in our everyday life. Experimental probability, with its basis in real-world observations, lends us the power to anticipate outcomes based on our experiences, paving the way for more informed decision-making.

From understanding the basic definition of probability to distinguishing between theoretical and experimental probability and mastering the formula of experimental probability, we hope you’re now well-equipped to navigate the captivating world of probability. Remember, probability isn’t just a concept confined within the pages of a mathematics textbook; it’s very much a part of the world around us, informing everything from weather forecasts to game strategy and risk analysis.

So, the next time you play a game of cards, shoot hoops, or even make a decision based on certain outcomes, remember the role of experimental probability! As always, the team at Brighterly is dedicated to making the learning of complex concepts enjoyable, ensuring you have fun on your journey of exploration. Stay tuned for more exciting math adventures!

Frequently Asked Questions on Experimental Probability

What is experimental probability.

Experimental probability is a probability value that is based on actual experiments or observations. In other words, it’s a type of probability that quantifies the ratio of the number of times an event occurs to the total number of trials or times an activity is performed. For example, if you flip a coin 100 times and it lands on heads 45 times, the experimental probability of getting heads is 45/100 = 0.45 or 45%.

How do you calculate experimental probability?

Calculating experimental probability is straightforward. It involves dividing the number of times an event occurs by the total number of trials. For instance, if you roll a die 60 times and get a ‘6’ on 10 occasions, the experimental probability of rolling a ‘6’ would be 10 (number of successful outcomes) divided by 60 (total number of outcomes), which equals 0.1667 or 16.67%.

What is the difference between theoretical and experimental probability?

Theoretical probability and experimental probability differ in their calculation and interpretation. Theoretical probability is a type of probability that assumes that all outcomes of an experiment are equally likely. It’s calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

On the other hand, experimental probability doesn’t rely on the assumption of equally likely outcomes but instead depends on actual data collected from conducted experiments. It deals with the frequency or proportion of times an event occurs based on experimental data.

Why is experimental probability important?

Experimental probability plays a crucial role in various fields and everyday life. Its importance lies in its basis on real-world data, which makes it a practical tool for predicting the likelihood of outcomes based on past experiences. Experimental probability is utilized in various sectors such as statistics, data analysis, gaming, weather forecasting, and in the medical field, among others. It also plays a key role in empirical research, where it aids in providing evidence-based conclusions.

• Wikipedia – Probability
• NCBI – Probability in Health
• Gov.uk – Understanding Uncertainty and Risk

As a seasoned educator with a Bachelor’s in Secondary Education and over three years of experience, I specialize in making mathematics accessible to students of all backgrounds through Brighterly. My expertise extends beyond teaching; I blog about innovative educational strategies and have a keen interest in child psychology and curriculum development. My approach is shaped by a belief in practical, real-life application of math, making learning both impactful and enjoyable.

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Lesson Explainer: Experimental Probability Mathematics

In this explainer, we will learn how to interpret a data set by finding and evaluating the experimental probability.

Probability refers to the likelihood or chance of an event occurring. In experimental probability, we make estimates for the likelihood or chance of something occurring based on the results of a number of experiments or trials.

Let us look at an example to illustrate what we mean by this. We will calculate the experimental probability using data in a table.

Example 1: Experimental Probability from a Table

The table shows the results of a survey that asked 20 students about their favorite breakfast.

What is the probability that a randomly selected student prefers eggs?

As we are using data collected in a survey, this is classified as experimental data. The 20 students asked about their favorite breakfast are the 20 trials in our experiment.

Since 10 out of the 20 students prefer eggs, the probability that a randomly selected student prefers eggs is p r o b a b i l i t y t h a t a s t u d e n t p r e f e r s e g g s E g g s n u m b e r o f s t u d e n t s w h o p r e f e r e g g s t o t a l n u m b e r o f s t u d e n t s = 𝑃 ( ) = = 1 0 2 0 = 1 2 = 0 . 5 .

The probability that a student selected at random prefers eggs for breakfast is therefore 0.5. Or converting this to a percentage, we can say there is a 50% chance that a student selected at random prefers eggs for breakfast. (Multiplying the probability 0.5 by 100 gives us the 50%.)

We will look at some more examples, but before we do this let us list the main points of interest for experimental probability.

Experimental Probability: Main Points

The experimental probability of event 𝐸 is an estimate of the probability for the event 𝑃 ( 𝐸 ) , based on data from a number of trials or experiments. So, for example, if we use data collected in a survey to estimate a probability, this would be classed as experimental probability.

• The experimental probability of an event is often also called the relative frequency of the event and is given by r e l a t i v e f r e q u e n c y o f e v e n t n u m b e r o f t i m e s o c c u r s t o t a l n u m b e r o f t r i a l s 𝐸 = 𝑃 ( 𝐸 ) = 𝐸 .
• As with any probability, if we have the experimental probability of an event 𝐸 , we can find the probability that 𝐸 does not occur, 𝑃 ( 𝐸 )  , by using the total probability rule: t h e s u m o f t h e p r o b a b i l i t i e s f o r a l l p o s s i b l e o u t c o m e s = 1 . So 𝑃 ( 𝐸 ) = 1 − 𝑃 ( 𝐸 )  .

The next example demonstrates how to calculate the relative frequency of an event.

Example 2: Experimental Probability and Relative Frequency

A coin was tossed 200 times and the number of tails observed was 102. Calculate the relative frequency of getting a heads. Calculate the answer to three decimal places.

As the coin was tossed 200 times, this means there were 200 “trials.” We are looking for the relative frequency of getting a heads, but we know that 102 of the throws resulted in tails. So to begin, we will calculate the relative frequency of getting a tails, which is r e l a t i v e f r e q u e n c y o f g e t t i n g a t a i l s T a i l s n u m b e r o f t i m e s t a i l s o c c u r r e d t o t a l n u m b e r o f t r i a l s = 𝑃 ( ) = = 1 0 2 2 0 0 .

Hence, the relative frequency of getting a tails is 1 0 2 2 0 0 . We can use this to find the relative frequency of getting a heads, which can be worked out in two different ways with the information we have.

• Method 1 uses the rule that the sum of the probabilities for all possible outcomes is equal to 1. We have just worked out the relative frequency (or probability) of getting a tails. This is 1 0 2 2 0 0 . Since there is only one other possible outcome (heads), subtracting the “tails” probability from 1 gives us the probability (or relative frequency) of getting a head: p r o b a b i l i t y o f h e a d s p r o b a b i l i t y o f t a i l s = 1 − = 1 − 1 0 2 2 0 0 = 9 8 2 0 0 . To three decimal places, 9 8 2 0 0 = 0 . 4 9 0 .
• Method 2 uses the number of tails that occurred and the total number of trials to calculate directly the number of heads out of all the trials. We then use this to calculate the relative frequency or probability of a heads: n u m b e r o f h e a d s n u m b e r o f t r i a l s n u m b e r o f t a i l s = − = 2 0 0 − 1 0 2 = 9 8 . As there were 98 heads out of 200 trials, the relative frequency of getting a heads is r e l a t i v e f r e q u e n c y o f h e a d s H e a d s n u m b e r o f t i m e s “ h e a d s ” o c c u r r e d t o t a l n u m b e r o f t r i a l s = 𝑃 ( ) = = 9 8 2 0 0 = 0 . 4 9 . The relative frequency of getting a heads is therefore 0.490 to 3 decimal places.

In our next example, we calculate the experimental probability for an event.

Example 3: Experimental Probability

A game at a festival challenged people to throw a baseball through a tire. Of the first 68 participants, 3 people won the gold prize, 12 won the silver prize, and 15 won the bronze prize.

What is the experimental probability of not winning any of the three prizes?

In order to find the experimental probability of not winning any of the three prizes, let us first summarize the information that we have:

The total number of trials is the total number of participants, which is 68. We know also that, of these 68 participants, 30 won a prize. If 30 won a prize, then 6 8 − 3 0 = 3 8 participants did not win a prize and we can use this information to calculate the experimental probability of not winning any of the three prizes: p r o b a b i l i t y o f n o t w i n n i n g a p r i z e n u m b e r o f n o n - p r i z e w i n n e r s t o t a l n u m b e r o f p a r t i c i p a n t s = = 3 8 6 8 = 1 9 3 4 .

Hence, the experimental probability of not winning a prize is 1 9 3 4 .

Note that we could also have used the total probability rule to answer this question (i.e., that the sum of the probabilities for all possible outcomes is equal to 1). With 68 participants, 30 of whom won a prize, the probability of winning a prize is 3 0 6 8 = 1 5 3 4 . By the total probability rule, the probability of not winning a prize is 1 − 𝑃 ( ) W i n n i n g . So p r o b a b i l i t y o f w i n n i n g ( p r o b a b i l i t y o f w i n n i n g ) a s b e f o r e n o t = 1 − = 1 − 1 5 3 4 = 1 9 3 4 .

Our next example demonstrates the use of tabular information in finding relative frequencies.

Example 4: Using Information from a Table to Find Relative Frequencies

The table shows the music preferences of a group of men and women.

• Calculate the relative frequency of a randomly selected person being a woman who prefers country music. If necessary, round your answer to 3 decimal places.
• Calculate the relative frequency of a randomly selected woman preferring rock music. If necessary, round your answer to 3 decimal places.

In order to find the relative frequencies, our first step is to calculate the total number of trials. We can work out the totals for each category as in the table below.

To calculate the relative frequency of a randomly selected person being a woman who prefers country music, we must find the number of people falling into this category and also the total number of people. (The total number of people is the number of trials.)

We can see that the number of women who prefer country music is 13 and that the total number of people whose preferences were recorded (i.e., the number of trials) is 63. The relative frequency (or probability 𝑃 ) of a randomly selected person being a woman (W) who prefers country music (C) is, therefore, 𝑃 ( ) = = 1 3 6 3 = 0 . 2 0 6 . W a n d C n u m b e r o f w o m e n w h o p r e f e r c o u n t r y m u s i c t o t a l n u m b e r o f p e o p l e t o 3 d . p

The relative frequency, to three decimal places, of randomly selecting a woman who prefers country music is therefore 0.206. Or as a percentage, there is a 20.6% ( = 0 . 2 0 6 × 1 0 0 ) chance that a person selected at random is a woman and prefers country music.

To calculate the relative frequency of a randomly selected woman preferring rock music, we need to know how many women there were in total and the number of those who prefer rock music.

There was a total of 37 women, 24 of whom prefer rock music. So the relative frequency of women who prefer rock music is r e l a t i v e f r e q u e n c y o f w o m e n w h o p r e f e r r o c k w o m e n r o c k e r s t o t a l w o m e n t o 3 d . p = = 2 4 3 7 = 0 . 6 4 9 .

Hence, the relative frequency of a randomly selected woman preferring rock music is 0.649 to three decimal places. Or, as a percentage, there is approximately a 65% ( ≈ 0 . 6 4 9 × 1 0 0 ) chance that a woman selected at random prefers rock music.

Note that there is a subtle difference between the wording of the two parts of this question. Part 1 asks for the relative frequency of a “randomly selected person being a woman who prefers country music.” And part 2 refers to a “randomly selected woman preferring rock music.”

The distinction is that, in part 2, we are making a random selection only from the women, whereas in part 1 we are randomly selecting from all of the people whose preferences were recorded. That is why in the solution to part 2 our denominator is the total number of women only, and in part 1 the denominator is the overall total, that is, both men and women.

Sometimes when looking at experimental probability, we may not be able to directly perform the experiment we would like but it is possible to model the situation. In the next example, we will consider how an experiment might be constructed to calculate experimental probability.

Example 5: Constructing an Experiment for Experimental Probability

One out of every six students in a seventh-grade class is left handed. Which of the following could be used to find the experimental probability that we will get a left-handed student when choosing randomly?

• Using a spinner with four colors: red represents right handed, and yellow, blue, and green represent left handed.
• Using a coin: heads is right handed, and tails is left handed.
• Using a number cube: even numbers represent right handed and odd numbers represent left handed.
• Using a number cube: landing on 1 represents left handed, and landing on 2–6 represents right handed.

Our information is that one in every six students is left handed. This means that for every individual left-handed student we expect there to be 5 right-handed students. So we are looking for an experiment with this ratio in its design. Let us consider each of the options separately and see if they fit the bill.

Let us remind ourselves of the key ideas related to experimental probability

• The experimental probability of event 𝐸 is an estimate of the probability for the event 𝑃 ( 𝐸 ) , based on data from a number of trials or experiments.
• The experimental probability of an event is also called the relative frequency of the event and is given by r e l a t i v e f r e q u e n c y o f e v e n t n u m b e r o f t i m e s o c c u r s t o t a l n u m b e r o f t r i a l s 𝐸 = 𝑃 ( 𝐸 ) = 𝐸 .
• When calculating probabilities, we often use the “total probability rule”: t h e s u m o f t h e p r o b a b i l i t i e s f o r a l l p o s s i b l e o u t c o m e s = 1 .

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• Experimental Probability - Definition And Examples

Experimental Probability Definition

Assume that a train is two hours late due to heavy weather, and that the train is scheduled to arrive at the station at 5:00 p.m. You are anticipating the arrival of the train at 5:05 p.m., which is an uncertain event. We can state the probability is less than or equal to one. The probability is the expectancy in this case.

The probability ranges from 0 to 1, with 0 indicating an impossible event and 1 indicating a certain event. It is the observational probability, also known as the empirical probability when the Experimental probability definition is described in experiments (or the relative frequency of events).

Theoretical And Experimental Probability

Theoretical probability assumes that everything will turn out perfectly. Assume you examined the weather for the past five days, beginning today. Today's forecast predicts rain for half of the day and clear skies for the rest. In the next four days, the same will be seen. If I predict that today would be 50% clear and 50% rainy, and assuming the best-case scenario, 70% of the next day will be clear and 30% will be rainy.

So, you simply made a hypothesis of the circumstance, which means you, when you assumed the rest to be exactly 50-50. The experimental probability was 70-30 when the result was 70-30. Because the experimental probability meaning is based on experiments, practical effort, or fieldwork rather than leading daydreaming assumptions, as you did in the example of the train's estimated arrival time.

As a result, the experimental probability gives you the precise outcome of an experiment, which may differ from the theoretical likelihood.

What Is Experimental Probability?

Experimental probability is a type of probability that is based primarily on a set of tests.

To evaluate their likelihood, a random experiment is conducted and repeated numerous times, with each repetition serving as a trial.

Because the experiment is being undertaken to determine whether or not an event will occur, i.e., the probability of an event occurring. Tossing a coin, throwing dice, or whirling a spinner are all examples. The probability of an event is always equal to the number of times it occurs divided by the total number of trials in mathematics.

Assume you flip a coin 50 times and keep track of whether you get a "head" or "tail." The experimental probability of getting a "tail" is computed as a percentage of the number of heads and total tosses, i.e.,

P (tail) = Number of tails recorded ÷  50 tosses

Where, P stands for the probability of an event occurring.

What does Experimental Probability Talk About?

The experimental probability describes the experiment's actual outcome. Let's imagine you run a 100-fold coin flip experiment. The coin has a theoretical probability of 50 percent heads and 50 percent tails.

In reality, the results of your experiment show 47 heads and 53 tails. This suggests that the experimental likelihood of receiving tails in 100 flips is 53 percent, whereas the experimental probability of getting heads in 100 trials is 47 percent. So, the 50-50 and 53-47 results, respectively, refer to theoretical and experimental probability.

Experimental Probability Formula

The experimental probability of an event occurring is calculated by dividing the number of times the event occurred during the experiment by the total number of times the experiment was conducted.

As a result, each possible outcome is uncertain, and the sample space is the collection of all possible outcomes. The Experimental Probability Formula assists us in calculating the experimental probability, which is calculated as follows:

Assume you spin a spinner 50 times, and the table below reveals the results of your experiment.

Image: 

We can now calculate the experimental probabilities of spinning the colour pink using this table.

Because a spinner turns 50 times and the pink colour appears 10 times, the total number of events or times a spinner revolves is 50.

As we know from the probability formula, the P(E) of an event is the number of occurrences divided by the total number of events done.

P (E)  =  10/50   = 1/5

As a result, the probability of the pink colour appearing on spinning is 1/5.

Let's look at some experimental probability examples to better comprehend the notion of experimental probability.

Experimental Probability Examples

1. The number of pancakes prepared by Fredrick per day this week is in the order of 4, 7, 6, 9, 5, 9, and 5. What will you say if I ask you to give me a credible estimate of the likelihood that Fredrick will make less than 6 pancakes the next day based on this data?

You say that P(< 6 pancakes) = 4, 5, 5 =  3 possibilities 

Mathematically, we get: 3/7 = 0.428 = 42%

As a result, there's a 42 percent chance that Fredrick will make less than six pancakes the next day.

2. Now you must calculate the likelihood that while ordering an exotica pizza, the next order will not include a Schezwan Sauce topping.

The following can be found on an exotica pizza:

Looking at the data table above, we can see that the realistic estimate of the probability that the next type of topping ordered will not be a Schezwan sauce is 27/43 = 62.8 %.

The preceding examples depict a real-life experimental probability scenario.

Experimental Probability Questions

1. The following table shows the observations made after throwing a 6-sided die 80 times:

Find the probability of an experiment in a throw of dice of a) obtaining a four; b) Obtaining a number less than 4, and c) Rolling a 3 or 6

We receive the numbers 1, 2, 3, 4, 5, and 6 from a single roll of the dice. Now we'll take each step toward our goals one by one. We know how to calculate the Experimental probability using the formula: The total number of trials divided by the number of times an event happens.

Obtaining a score of 4: 14/80=0.175, or 17.5 percent.

When you roll a number that is fewer than four, you have a chance of winning. One, two, and three are the outcomes. Each has a frequency of 13, 10, and 15 respectively. Now add them all together to obtain the total number of times an event happened, which is 38. P (numbers less than 4) = 38/80 = 0.475 or 47.5 percent.

The same goes for rolling a 3 or 6: 31/80 = 0.387 or 38.7%.

Q2: Which of the following is a probability experiment?

Answer: Option (d): The value of experimental probability represented as a percentage ranges from 0 to 1.

Practise Question MCQs

1. If the probability of an event happening is 0.3 and the probability of the event not happening is_____

None of the above

2. 200 times, three coins were tossed. There were 72 times when two heads appeared.  Then the probability of 2 heads coming up is

In the nutshell, the experimental probability focuses on the result of an experiment, while the theoretical probability is just an assumption that we make to work on our experiments. 

FAQs on Experimental Probability - Definition And Examples

1. What are the important points to be kept while dealing with probability?

The following are the key aspects to remember when learning about experimental probability:

The sum of all experimental probabilities for all outcomes is always one.

An unclear event's probability ranges from 0 to 1, with 0 denoting an impossible occurrence and 1 denoting a certain event.

The likelihood can be expressed as a percentage.

2. The following table shows the number of offers Mike received lately when shopping at seven different malls. 4, 3, 2, 1, 6, 8, and 9 are the numbers. Determine the likelihood that Mike will receive no offers from two of the seven malls on his next shopping trip.

The probability that mike received no offer from two of the seven malls in the next shopping is given by:

P(E) =  P(<  2) = 2/7 or 28.57%.

3. What are the three types of probability?

The following are the three forms of probability:

Theoretical probability, Axiomatic probability and experimental probability.

4. What happens to experimental probability when the number of trials increases?

When we increase the number of trials of flipping a coin or tossing dice in experimental probability, we discover that the experimental probability approaches the theoretical probability.

Experimental Probability

Related Topics: More Lessons for Probability Math Worksheets

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• Experimental Probability

Experimental probability , also known as empirical probability , is a concept in mathematics that deals with estimating the likelihood of an event occurring based on actual experimental results. Unlike theoretical probability , which predicts outcomes based on known possibilities, experimental probability is derived from real-life experiments and observations.

To understand this better, imagine flipping a coin. The theoretical probability of landing heads is 50% or 1/2. However, if you actually flip the coin 100 times and record the outcomes, you might get heads 48 times. The experimental probability of getting heads would then be 48/100 or 0.48.

In this article, we will explore the concept of experimental probability, its significance, and how it differs from theoretical probability. We will discuss the formula for calculating experimental probability, provide examples to illustrate its application.

Table of Content

What is Probability?

What is experimental probability, formula for experimental probability, examples of experimental probability, what is theoretical probability, experimental probability vs theoretical probability.

• Solved Examples
• Practice Problems

The branch of mathematics that tells us about the likelihood of the occurrence of any event is the probability . Probability tells us about the chances of happening an event.

The probability of any element that is sure to occur is One(1) whereas the probability of any impossible event is Zero(0). The probability of all the elements ranges between 0 to 1.

There are two ways of studying probability that are

• Theoretical Probability

Now let’s learn about both in detail.

Experimental probability is a type of probability that is calculated by conducting an actual experiment or by performing a series of trials to observe the occurrence of an event. It is also known as empirical probability.

To calculate experimental probability, you need to conduct an experiment by repeating the event multiple times and observing the outcomes. Then, you can find the probability of the event occurring by dividing the number of times the event occurred by the total number of trials.

The experimental Probability for Event A can be calculated as follows:

P(E) = (Number of times an event occur in an experiment) / (Total number of Trials)

Now, as we learn the formula, let’s put this formula in our coin-tossing case.  If we tossed a coin 10 times and recorded a head 4 times and a tail 6 times then the Probability of Occurrence of Head on tossing a coin:

P(H) = 4/10

Similarly, the Probability of Occurrence of Tails on tossing a coin:

P(T) = 6/10

Theoretical Probability deals with assumptions in order to avoid unfeasible or expensive repetition experiments. The theoretical Probability for an Event A can be calculated as follows:

P(A) = Number of outcomes favorable to Event A / Number of all possible outcomes            

Now, as we learn the formula, let’s put this formula in our coin-tossing case. In tossing a coin, there are two outcomes: Head or Tail.

Hence, The Probability of occurrence of Head on tossing a coin is

Similarly, The Probability of the occurrence of a Tail on tossing a coin is

There are some key differences between Experimental and Theoretical Probability , some of which are as follows:

• Probability in Maths
• Probability Distribution
• Bayes’ Theorem

Solved Examples of Experimental Probability

Example 1. Let’s take an example of tossing a coin, tossing it 40 times , and recording the observations. By using the formula, we can find the experimental probability for heads and tails as shown in the below table.

Number of Trail Outcome Number of Trail Outcome Number of Trail Outcome Number of Trail Outcome First H Eleventh T Twenty-first T Thirty-first T Second T Twelfth T Twenty-second H Thirty-second H Third T Thirteenth H Twenty-third T Thirty-third T Fourth H Fourteenth H Twenty-fourth H Thirty-fourth H Fifth H Fifteenth H Twenty-fifth T Thirty-fifth T Sixth H Sixteenth H Twenty-sixth H Thirty-sixth T Seventh T Seventeenth T Twenty-seventh T Thirty-seventh T Eighth H Eighteenth T Twenty-eighth T Thirty-eighth H Ninth T Nineteenth T Twenty-ninth T Thirty-ninth T Tenth H Twentieth T Thirtieth H Fortieth T The formula for experimental probability: P(H) = Number of Heads ÷ Total Number of Trials = 16 ÷ 40 = 0.4 Similarly, P(H) = Number of Tails ÷ Total Number of Trials = 24 ÷ 40 = 0.6 P(H) + P(T) = 0.6 + 0.4 = 1 Note: Repeat this experiment for ‘n’ times and then you will find that the number of times increases, the fraction of experimental probability comes closer to 0.5. Thus if we add P(H) and P(T), we will get  0.6 + 0.4 = 1 which means P(H) and P(T) is the only possible outcomes.

Example 2. A manufacturer makes 50,000 cell phones every month. After inspecting 1000 phones, the manufacturer found that 30 phones are defective. What is the probability that you will buy a phone that is defective? Predict how many phones will be defective next month.

Experimental Probability = 30/1000 = 0.03 0.03 = (3/100) × 100 = 3% The probability that you will buy a defective phone is 3% ⇒ Number of defective phones next month = 3% × 50000 ⇒ Number of defective phones next month = 0.03 × 50000 ⇒ Number of defective phones next month = 1500

Example 3. There are about 320 million people living in the USA. Pretend that a survey of 1 million people revealed that 300,000 people think that all cars should be electric. What is the probability that someone chosen randomly does not like the electric car? How many people like electric cars?

Now the number of people who do not like electric cars is 1000000 – 300000 = 700000 Experimental Probability =  700000/1000000 = 0.7 And, 0.7 = (7/10) × 100 = 70% The probability that someone chose randomly does not like the electric car is 70% The probability that someone like electric cars is  300000/1000000 = 0.3 Let x be the number of people who love electric cars ⇒ x = 0.3 × 320 million ⇒ x = 96 million The number of people who love electric cars is 96 million.

Practice Problems on Experimental Probability

Problem 1: A coin is flipped 200 times, and it lands on heads 120 times. What is the experimental probability of getting heads?

Problem 2: A die is rolled 50 times, and the number 3 appears 8 times. What is the experimental probability of rolling a 3?

Problem 3: In a class survey, 150 students were asked if they prefer reading books or watching movies. 90 students said they prefer watching movies. What is the experimental probability that a randomly chosen student prefers watching movies?

Problem 4: A bag contains 5 red, 7 blue, and 8 green marbles. If 40 marbles are drawn at random with replacement, and 12 of them are red, what is the experimental probability of drawing a red marble?

Problem 5: A basketball player made 45 successful free throws out of 60 attempts. What is the experimental probability that the player will make a free throw?

Problem 6: During a game, a spinner is spun 80 times, landing on a specific section 20 times. What is the experimental probability of the spinner landing on that section?

FAQs on Experimental Probability

Define experimental probability..

Probability of an event based on an actual trail in physical world is called experimental probability.

How is Experimental Probability calculated?

Experimental Probability is calculated using the following formula:  P(E) = (Number of trials taken in which event A happened) / Total number of trials

Can Experimental Probability be used to predict future outcomes?

No,  experimental probability can’t be used to predict future outcomes as it only achives the theorectical value when the trails becomes infinity.

How is Experimental Probability different from Theoretical Probability?

 Theoretical probability is the probability of an event based on mathematical calculations and assumptions, whereas experimental probability is based on actual experiments or trials.

What are some Limitations of Experimental Probability?

There are some limitation of experimental probability, which are as follows: Experimental probability can be influenced by various factors, such as the sample size, the selection process, and the conditions of the experiment.  The number of trials conducted may not be sufficient to establish a reliable pattern, and the results may be subject to random variation.  Experimental probability is also limited to the specific conditions of the experiment and may not be applicable in other contexts.

Can Experimental Probability of an event be a negative number if not why?

As experimental probability is given by: P(E) = Number of trials taken in which event A happened/Total number of trials Thus, it can’t be negative as both number are count of something and counting numbers are 1, 2, 3, 4, …. and they are never negative.

What are Types of Probability?

There are two forms of calculating the probability of an event that are, Theoretical Probability Experimental Probability

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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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Probability

How likely something is to happen.

Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Tossing a Coin

When a coin is tossed, there are two possible outcomes:

Heads (H) or Tails (T)

• the probability of the coin landing H is ½
• the probability of the coin landing T is ½

Throwing Dice

When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6 .

The probability of any one of them is 1 6

In general:

Probability of an event happening = Number of ways it can happen Total number of outcomes

Example: the chances of rolling a "4" with a die

Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability = 1 6

Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

Number of ways it can happen: 4 (there are 4 blues)

Total number of outcomes: 5 (there are 5 marbles in total)

So the probability = 4 5 = 0.8

Probability Line

We can show probability on a Probability Line :

Probability is always between 0 and 1

Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide

Example: toss a coin 100 times, how many Heads will come up?

Probability says that heads have a ½ chance, so we can expect 50 Heads .

But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.

Learn more at Probability Index .

Some words have special meaning in Probability:

Experiment : a repeatable procedure with a set of possible results.

Example: Throwing dice

We can throw the dice again and again, so it is repeatable.

The set of possible results from any single throw is {1, 2, 3, 4, 5, 6}

Outcome: A possible result.

Example: "6" is one of the outcomes of a throw of a die.

Trial: A single performance of an experiment.

Example: I conducted a coin toss experiment. After 4 trials I got these results:

Three trials had the outcome "Head", and one trial had the outcome "Tail"

Sample Space: all the possible outcomes of an experiment.

Example: choosing a card from a deck

There are 52 cards in a deck (not including Jokers)

So the Sample Space is all 52 possible cards : {Ace of Hearts, 2 of Hearts, etc... }

The Sample Space is made up of Sample Points:

Sample Point: just one of the possible outcomes

Example: Deck of Cards

• the 5 of Clubs is a sample point
• the King of Hearts is a sample point

"King" is not a sample point. There are 4 Kings, so that is 4 different sample points.

There are 6 different sample points in that sample space.

Event: one or more outcomes of an experiment

Example Events:

An event can be just one outcome:

• Getting a Tail when tossing a coin
• Rolling a "5"

An event can include more than one outcome:

• Choosing a "King" from a deck of cards (any of the 4 Kings)
• Rolling an "even number" (2, 4 or 6)

Hey, let's use those words, so you get used to them:

Example: Alex wants to see how many times a "double" comes up when throwing 2 dice.

The Sample Space is all possible Outcomes (36 Sample Points):

{1,1} {1,2} {1,3} {1,4} ... ... ... {6,3} {6,4} {6,5} {6,6}

The Event Alex is looking for is a "double", where both dice have the same number. It is made up of these 6 Sample Points :

{1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}

These are Alex's Results:

 After 100 Trials , Alex has 19 "double" Events ... is that close to what you would expect?