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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

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.

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

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

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

Calculating the probability of an event is determining the likelihood that this event will occur. There are two main ways in which we can estimate the probability of an event. One way is to consider the attributes or physical properties of the event in question. For example, if we wanted to calculate the probability of rolling a 5 on a fair die, we would consider the number of sides on the die. This probability, 1 6 , would be the theoretical probability of rolling a 5.

Often, however, we cannot use theoretical probability, for example, if we wanted to determine the probability of rolling a 5 on an unfair die. In this case, we would need to carry out an experiment where we roll the die a number of times, for example, 100 times, and record the number of times a 5 was rolled. This is termed experimental probability . To gather data to calculate experimental probability, we perform repeated trials and record the outcome (the result) of each trial.

Definition: Experimental Probability

• The experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials in an experiment. E x p e r i m e n t a l p r o b a b i l i t y n u m b e r o f t r i a l s i n w h i c h t h e o u t c o m e 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 = .

Let’s consider an example where we have a biased (unfair) spinner with values from 1 to 6 that has some of its sides weighted. We want to determine the probability of the spinner landing on 2.

As the spinner is biased, we cannot use the fact that it has 6 equal sides to determine the probability of getting any one of the values. We would need to perform an experiment where we repeatedly spin the spinner a number of times and record the outcome of each spin. A tally chart or table is useful for recording these results. The more trials that we perform, the more reliable our results will be. However, we must also balance this with the practical considerations of time and cost.

Let’s say that after the experiment is complete, we have the table of outcomes below.

There are a number of things we can observe from the table, for instance, the value 3 on the spinner occurred most times: it had a frequency of 25 spins. In comparison, the value 6 was never spun on the spinner during the experiment: its frequency was 0.

From a table such as this, we can also determine the number of trials that were carried out in the experiment: we would calculate the total number of outcomes. This would give t o t a l n u m b e r o f t r i a l s = 2 + 1 6 + 2 5 + 1 7 + 4 + 0 = 6 4 .

Importantly, we can also use the results of this experiment to find the experimental probability of spinning each or any value on the spinner. In general, to calculate experimental probability, we have e x p e r i m e n t a l p r o b a b i l i t y n u m b e r o f t r i a l s i n w h i c h t h e o u t c o m e 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 = .

We can reword this formula to match the context of the probability that we need to calculate. Here, we could write the experimental probability of spinning a 2 as e x p e r i m e n t a l p r o b a b i l i t y o f s p i n n i n g a n u m b e r o f t r i a l s i n w h i c h 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 2 = 2 .

Given that the number of trials in which 2 occurs is 16 and the total number of trials is 60, we have e x p e r i m e n t a l p r o b a b i l i t y o f s p i n n i n g a 2 = 1 6 6 4 = 1 4 .

We can write probabilities as fractions, decimals, or percentages, so an answer of 1 4 , 0.25, or 2 5 % would be valid.

Note that in a context where we are asked to find the probability of an event from data given as a set of results, a table, or a graph, this will be the experimental probability.

Let’s now see how we can apply this in the following examples.

Example 1: Calculating the Experimental Probability of an Event Using a Frequency 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?

In the table above, we can see that out of 20 students, 10 students said that eggs were their favorite breakfast, 2 students said that cereal was their favorite, and 8 students said that toast was their favorite. In order to calculate the probability that a student prefers eggs, we will calculate the experimental probability of this. In general, we use the formula e x p e r i m e n t a l p r o b a b i l i t y n u m b e r o f t r i a l s i n w h i c h t h e o u t c o m e 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 = .

In this context, the experimental probability of a randomly selected student preferring eggs can be written as p r o b a b i l i t y t h a t e g g s a r e p r e f e r r e d 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 r e d 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 s u r v e y e d = .

From the table, we have the number of students surveyed who preferred eggs as 10, and we are given that the total number of students surveyed is 20. Even if we did not have the information about the total number of students being 20, we could calculate this by summing the number of students preferring eggs (10), cereal (2), and toast (8).

Hence, we have p r o b a b i l i t y t h a t e g g s a r e p r e f e r r e d = 1 0 2 0 = 1 2 .

Therefore, we can give the answer that the probability that a randomly selected student prefers eggs is 1 2 .

Let’s see another example where the outcomes of an experiment are given as a list, or set, of results.

Example 2: Calculating the Experimental Probability of an Event Using a Set

Dina creates a three-sided spinner using the colors red, green, and blue. She spins the spinner and records the following results:

{red, blue, red, green, green, green, red, red, red, green}.

Calculate the experimental probability of spinning green on this spinner.

The experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials in an experiment. It can be written as e x p e r i m e n t a l p r o b a b i l i t y n u m b e r o f t r i a l s i n w h i c h t h e o u t c o m e 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 = .

In this problem, we will calculate e x p e r i m e n t a l p r o b a b i l i t y o f s p i n n i n g g r e e n n u m b e r o f t i m e s g r e e n w a s s p u n t o t a l n u m b e r o f s p i n s = .

Using the given set of results, we can determine that green was spun 4 times, as it appears 4 times in the list of results. By counting all the results, we determine that the total number of spins in the experiment is 10, which is the size of the set of results. Thus, we have e x p e r i m e n t a l p r o b a b i l i t y o f s p i n n i n g g r e e n n u m b e r o f t i m e s g r e e n w a s s p u n t o t a l n u m b e r o f s p i n s = = 4 1 0 = 2 5 .

Probability can be given as a fraction, decimal, or percentage, so 2 5 , 0.4, or 4 0 % are valid answers here.

We can give the answer that the experimental probability of spinning green on this spinner is 2 5 .

Note that in this question, the spinner was spun 10 times. Ideally, a larger number of spins would give a more accurate result.

Experimental probability is particularly useful in allowing us to determine the probability of an event happening on a wider scale by performing an experiment on a smaller scale. This is often done in industry by carrying out experiments on a sample. For example, a manufacturer of light bulbs might test a sample of light bulbs to determine the length of their lifetime. Since this would destroy the bulbs that are tested, the manufacturer cannot perform this check on each of the bulbs the manufacturer has produced. Rather, selecting 100 light bulbs and determining that 97 of them reached a target time of 500 hours would allow the manufacturer to extrapolate that 0.97 of the manufactured bulbs have a lifetime of at least 500 hours .

We will now see an example of finding the experimental probability of an event from a sample.

Example 3: Estimating the Probability of an Event Using Experimental Data

A store receives a box of apples from an orchard. A worker inspects a sample of 54 apples from the box. Of these apples, 6 are spoiled. Use this data to estimate the probability that an apple received from the orchard is spoiled.

We can estimate the probability of the store receiving a spoiled apple by applying experimental probability. In this context, the experiment is that of repeated trials of selecting apples. The two different outcomes are selecting a spoiled apple or selecting an unspoiled apple.

We can calculate the probability of selecting a spoiled apple using the number of spoiled apples (6) and the total number of apples (54) as p r o b a b i l i t y o f a s p o i l e d a p p l e n u m b e r o f s p o i l e d a p p l e s t o t a l n u m b e r o f a p p l e s = = 6 5 4 = 1 9 .

Hence, we can give the answer that the estimated probability of receiving a spoiled apple is 1 9 .

The way in which the data from experimental probability is presented can be in the form of a statement, a set of results, a table, or a graph. We still apply the same process of calculating the experimental probability of an event by dividing the number of trials in which that particular outcome occurs by the total number of trials. In many cases, we may need to sum the frequencies of the different outcomes to find the total number of trials.

In the following question, we will see how we can use a bar graph to calculate an experimental probability.

Example 4: Calculating the Experimental Probability of an Event Using a Graph

The graph shows the results of an experiment in which a die was rolled 26 times. Find the experimental probability of rolling a 2. Give your answer as a fraction in its simplest form.

In this question, we are given a bar graph representing the number of times the values 1 to 6 were rolled on a die. We can calculate the experimental probability of rolling a 2 as p r o b a b i l i t y o f r o l l i n g a n u m b e r o f t i m e s w a s r o l l e d t o t a l n u m b e r o f r o l l s 2 = 2 .

Using the bar graph, we can observe that the number of times 2 was rolled is 8.

We are given that the total number of rolls is 26. We can verify this by finding the total of all the frequencies (the total number of rolls) of the values 1 to 6. This is given by 4 + 8 + 8 + 3 + 1 + 2 = 2 6 . Note that we must also include the number of times 2 was rolled (8) in our sum.

We now calculate the experimental probability of rolling a 2 as e x p e r i m e n t a l p r o b a b i l i t y o f r o l l i n g a 2 = 8 2 6 = 4 1 3 .

Therefore, the experimental probability of rolling a 2 is 4 1 3 .

We will now see another question.

Example 5: Calculating the Experimental Probability of an Event

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 this game, we can observe that there are, in fact, 4 different outcomes: winning gold, silver, bronze, or not winning any of these prizes.

Given the information that 3 people won the gold prize, 12 won the silver prize, and 15 won the bronze prize out of a total of 68 participants, we can calculate the number of participants who won no prizes as n u m b e r o f p a r t i c i p a n t s w h o w o n n o p r i z e s = 6 8 − ( 3 + 1 2 + 1 5 ) = 6 8 − 3 0 = 3 8 .

We can then calculate the experimental probability of not winning any prizes by dividing the number of participants who won no prizes by the total number of participants, that is, e x p e r i m e n t a l p r o b a b i l i t y o f n o t w i n n i n g a n y p r i z e s n u m b e r o f p a r t i c i p a n t s w h o w o n n o p r i z e 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 = .

Thus, we have e x p e r i m e n t a l p r o b a b i l i t y o f n o t w i n n i n g a n y p r i z e s = 3 8 6 8 = 1 9 3 4 .

Therefore, we can give the answer that the experimental probability of not winning any prizes in this game is 1 9 3 4 .

In the final question, we will use a given experimental probability and a value for the number of outcomes to calculate the total number of trials in an experiment.

Example 6: Determining the Total Number of Trials in an Experiment

The experimental probability that a coin lands on tails is 3 7 . If the coin landed on tails 30 times, how many times was it tossed in the experiment?

In this question, we are given the experimental probability that a coin lands on tails. This probability has been calculated using the data from an experiment with repeated trials of tossing a coin. We are also given that the number of outcomes of landing on tails (the number of times the coin landed on tails) is 30. We recall that in general, experimental probability is calculated as e x p e r i m e n t a l p r o b a b i l i t y n u m b e r o f t r i a l s i n w h i c h t h e o u t c o m e 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 = .

In this context, we would have p r o b a b i l i t y o f a c o i n l a n d i n g o n t a i l s n u m b e r o f t i m e s t h e c o i n l a n d e d o n t a i l s t o t a l n u m b e r o f t i m e s t h e c o i n w a s t o s s e d = .

Given that the probability of the coin landing on tails is 3 7 and the coin landed on tails 30 times, we have 3 7 = 3 0 . t o t a l n u m b e r o f t i m e s t h e c o i n w a s t o s s e d

Then, by cross multiplying, we have 3 × ( ) = 3 0 × 7 . t o t a l n u m b e r o f t i m e s t h e c o i n w a s t o s s e d

Dividing both sides by 3 and simplifying gives us t o t a l n u m b e r o f t i m e s t h e c o i n w a s t o s s e d = 2 1 0 3 = 7 0 .

Therefore, we can give the answer that the coin was tossed 70 times.

We now summarize the key points of this explainer.

• Experiments can be used to estimate the probability of an event occurring.
• To gather data to calculate experimental probability, we perform repeated trials and record the outcome (the result) of each trial.
• The more trials we perform, the more accurate an estimate of results we will get. However, in a real-life context, we must balance this with the time and cost considerations of performing a large number of trials.
• We can calculate experimental probability from data presented in the form of a statement, set of results, a table, or a graph.

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• Math Article
• Experimental Probability

You and your 3 friends are playing a board game. It’s your turn to roll the die and to win the game you need a 5 on the dice. Now, is it possible that upon rolling the die you will get an exact 5? No, it is a matter of chance. We face multiple situations in real life where we have to take a chance or risk. Based on certain conditions, the chance of occurrence of a certain event can be easily predicted. In our day to day life, we are more familiar with the word ‘ chance and probability ’. In simple words, the chance of occurrence of a particular event is what we study in probability. In this article, we are going to discuss one of the types of probability called  “Experimental Probability” in detail.

What is Probability?

Probability, a branch of Math that deals with the likelihood of the occurrences of the given event. The probability values for the given experiment is usually defined between the range of numbers. The values lie between the numbers 0 and 1. The probability value cannot be a negative value. The basic rules such as addition, multiplication and complement rules are associated with the probability.

Experimental Probability Vs Theoretical Probability

There are two approaches to study probability:

• Theoretical Probability

What is Experimental Probability?

Experimental probability, also known as Empirical probability, is based on actual experiments and adequate recordings of the happening of events. To determine the occurrence of any event, a series of actual experiments are conducted. Experiments which do not have a fixed result are known as random experiments. The outcome of such experiments is uncertain. Random experiments are repeated multiple times to determine their likelihood. An experiment is repeated a fixed number of times and each repetition is known as a trial. Mathematically, the formula for the experimental probability is defined by;

Probability of an Event P(E) = Number of times an event occurs / Total number of trials.

What is Theoretical Probability?

In probability, the theoretical probability is used to find the probability of an event. Theoretical probability does not require any experiments to conduct. Instead of that, we should know about the situation to find the probability of an event occurring. Mathematically, the theoretical probability is described as the number of favourable outcomes divided by the number of possible outcomes.

Probability of Event P(E) = No. of. Favourable outcomes/ No. of. Possible outcomes.

Experimental Probability Example

Example: You asked your 3 friends Shakshi, Shreya and Ravi to toss a fair coin 15 times each in a row and the outcome of this experiment is given as below:

Calculate the probability of occurrence of heads and tails.

Solution: The experimental probability for the occurrence of heads and tails in this experiment can be calculated as:

Experimental Probability of Occurrence of heads = Number of times head occurs/Number of times coin is tossed.

Experimental Probability of Occurrence of tails = Number of times tails occurs/Number of times coin is tossed.

We observe that if the number of tosses of the coin increases then the probability of occurrence of heads or tails also approaches to 0.5.

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Experimental Probability – Definition with Examples

Updated on January 9, 2024

At Brighterly , we believe that a solid understanding of mathematics can empower our children to do great things. That’s why we’re committed to making complex math concepts accessible, engaging, and fun for all children. Among the myriad of mathematical topics we cover, one of the more practical, yet fascinating, is experimental probability.

Experimental probability is like the bridge between math and the real world, offering a hands-on approach to understanding likelihood and chance. It’s all about observation, data collection, and making sense of the patterns that emerge. Experimental probability takes us beyond the theoretical and into the empirical, providing our children with a richer, fuller understanding of how probability works.

What Is Experimental Probability?

Experimental probability is a concept that children often encounter in their mathematical journey, and it provides a fantastic way to understand how probability works in the real world. It is a type of probability that we calculate based on the outcomes of an experiment or activity, as opposed to theoretical probability which we calculate using mathematical principles. It’s all about doing rather than just thinking.

Imagine you’re flipping a coin. The theoretical probability of getting a heads or tails is 50%, or 0.5, because these are the only two possible outcomes. However, if you flip the coin 10 times and get 7 heads and 3 tails, the experimental probability of getting heads is 70% (or 0.7), and for tails, it’s 30% (or 0.3). This is because experimental probability depends on the actual results of the experiment.

Definition of Experimental Probability

The definition of experimental probability is the ratio of the number of times an event occurs to the total number of trials or times the activity is performed. It is calculated after conducting an experiment or activity, and can often differ from theoretical probability because of the variability and unpredictability of real-world events.

Calculating Experimental Probability

The process of calculating experimental probability involves two steps: conducting the experiment to gather data, and then using that data to calculate the probability. The formula for calculating experimental probability is:

P(E) = Number of times event E occurs / Total number of trials

For example, if you roll a dice 60 times, and the number 4 comes up 15 times, the experimental probability of rolling a 4 is calculated as 15 (the number of times 4 occurs) divided by 60 (the total number of trials), which equals 0.25, or 25%.

Examples of Experimental Probability

To better understand this concept, let’s explore some real-world examples of experimental probability:

In a bag of 30 marbles, 10 are blue, 10 are green, and 10 are red. If you randomly pick a marble 30 times, replacing the marble each time, and you get 12 blue, 8 green, and 10 red marbles, the experimental probability for each color would be calculated as follows:

• Blue: 12/30 = 0.4 or 40%
• Green: 8/30 = 0.267 or 26.7%
• Red: 10/30 = 0.333 or 33.3%

In a deck of 52 playing cards, if you draw a card 52 times, replacing the card each time, and you draw a heart 15 times, the experimental probability of drawing a heart is 15/52 = 0.288 or 28.8%.

Properties of Experimental Probability

Experimental probability, as with any type of probability, possesses some key properties. It will always be a value between 0 and 1 (or 0% and 100% when expressed as a percentage). This makes sense, as it’s impossible for an event to occur less than 0 times (probability = 0), or more times than the total number of trials (probability = 1).

Another key property is that the sum of the probabilities of all possible outcomes will equal 1. For example, in our earlier coin flipping example, the sum of the experimental probabilities for getting heads (0.7) and tails (0.3) equals 1.

Key Factors Affecting Experimental Probability

The key factor affecting experimental probability is the number of trials. In general, the more trials are performed, the closer the experimental probability gets to the theoretical probability. This principle is known as the Law of Large Numbers.

Other factors that can affect experimental probability include inaccuracies in data collection and environmental variables, such as the fairness of a coin or die, the method of drawing cards, and so on.

Difference Between Experimental and Theoretical Probability

The main difference between experimental and theoretical probability lies in their calculation methods. Theoretical probability is determined mathematically, using the known outcomes of an event, while experimental probability is determined empirically, using data from actual trials of the event.

In theory, a coin has a 50% chance of landing on heads, but in an experiment, it might not. Over the long run, the experimental probability will likely get closer to the theoretical probability, thanks to the Law of Large Numbers.

Formulas for Calculating Experimental Probability

As mentioned earlier, the formula for calculating experimental probability is straightforward:

Here, ‘P(E)’ represents the probability of event E occurring.

Writing Formulas for Experimental Probability

Let’s get into the details of how to write formulas for experimental probability. For any given event E, you can express the experimental probability of that event occurring as a fraction, decimal, or percentage using the aforementioned formula. Just remember to divide the number of times the event occurred by the total number of trials.

For example, if you’re trying to find the experimental probability of drawing a heart from a deck of cards and you drew a heart 13 times out of 52 trials, you’d write it as follows:

P(Heart) = 13/52 ≈ 0.25 = 25%

Use Cases of Experimental Probability in Real Life

Experimental probability finds its use in various real-life scenarios, from games and sports to weather forecasting and medical research. For example, predicting the outcome of a football game based on past performances is a use of experimental probability. Likewise, weather forecasts use data from previous years to predict the likelihood of certain weather conditions. Experimental probability is also used in clinical trials to determine the effectiveness of a new drug or treatment.

Practice Problems on Experimental Probability

To fully understand experimental probability, it’s helpful to solve some practice problems. Try the following scenarios:

• You toss a coin 50 times and get heads 29 times. What is the experimental probability of getting heads?
• You draw a card from a deck of 52 cards 100 times and draw a queen 22 times. What is the experimental probability of drawing a queen?
• You roll a die 200 times and roll a 5, 40 times. What is the experimental probability of rolling a 5?

In conclusion, experimental probability offers a practical and exciting way for children to understand the concept of probability and chance. Through experiments and observations, children can learn not just how to calculate the likelihood of an event, but also develop an intuitive understanding of probability.

At Brighterly, we encourage our young learners to immerse themselves in the world of experimental probability and explore its numerous applications in real-life situations. From games and sports to weather forecasting and medical research, experimental probability has vast real-world significance. Remember, with more trials, the experimental probability tends to converge with the theoretical probability, making it a valuable tool in understanding uncertainty and making predictions.

Frequently Asked Questions on Experimental Probability

What is the formula for experimental probability.

The formula for experimental probability is: P(E) = Number of times event E occurs / Total number of trials. Here, P(E) stands for the probability of event E, which could be any event you’re examining. This formula is straightforward to use, and it allows you to compute the experimental probability accurately using your collected data.

How is experimental probability calculated?

Experimental probability is calculated by carrying out an experiment and recording the outcomes. The number of times a particular event occurs is then divided by the total number of trials conducted. For example, if you roll a dice 100 times and the number 4 comes up 20 times, the experimental probability of rolling a 4 would be 20/100 = 0.20, or 20%.

What is the difference between experimental and theoretical probability?

Theoretical probability and experimental probability differ in the ways they are determined. Theoretical probability is derived using mathematical principles, considering all possible outcomes of an event. For instance, when flipping a fair coin, the theoretical probability of getting a head is 50% since there are two equally likely outcomes – heads and tails. On the other hand, experimental probability is calculated based on actual experiments or trials. If you flip the same coin 100 times and get heads 60 times, the experimental probability of getting heads would be 60/100 = 0.60, or 60%. Over time, with a large number of trials, the experimental probability will tend to get closer to the theoretical probability. This is a consequence of the Law of Large Numbers.

• Britannica: Law of Large Numbers
• Coursera: Understanding Experimental Probability
• Wolfram MathWorld: Experimental Probability

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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How To Calculate Experimental Probability: Step-By-Step Guide With Examples

Experimental probability.

Experimental probability is an approach to calculating the likelihood of an event occurring based on the results of an experiment or trial. This method involves conducting multiple trials or tests of an experiment and then calculating the probability of an event occurring based on the frequency of its occurrence in those trials.

Here are the steps involved in determining experimental probability:

1. Define the event: Start by defining the event for which you want to calculate the probability. For example, if you are flipping a coin, the event could be getting heads.

2. Conduct the experiment: Carry out the experiment by flipping the coin a predetermined number of times. For example, if you want to flip the coin 10 times, then do so and record the results.

3. Count the number of occurrences: Count the number of times the event occurred during the experiment. For example, if you flipped the coin 10 times and got heads 4 of those times, then the number of occurrences of the event (heads) is 4.

4. Calculate the experimental probability: Determine the experimental probability by dividing the number of occurrences by the total number of trials. In this example, the experimental probability of getting heads is 4/10, or 0.4 (or 40%).

5. Repeat and refine: To increase the accuracy of your results, continue repeating the experiment multiple times and refining your calculations.

In summary, experimental probability involves conducting an experiment or trial, counting the number of times an event occurs, and then calculating the probability of the event based on the frequency of its occurrence.

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Experimental Probability

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Experimental probability refers to the probability of an event based on actual experimentation or observation of outcomes.

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Experimental probability refers to the probability of an event based on actual experimentation or observation of outcomes. It is determined by conducting an experiment or observing an event multiple times and recording the number of times the event occurs.

To find the experimental probability of an event, you would divide the number of times the event occurred by the total number of trials or observations. For example, if you flipped a coin 20 times and it landed on heads 12 times, the experimental probability of flipping heads would be 12/20 or 0.6.

Experimental probability is often used in situations where it is difficult or impossible to determine the theoretical probability of an event. It can be used to estimate the theoretical probability, but it may not be as accurate as using mathematical formulas to calculate probability.

However, experimental probability can still provide valuable information about the likelihood of an event occurring, especially if the sample size is large enough to reduce the effects of randomness and variability.

Experimental Probability Examples:

Example 1: You roll a six-sided die 100 times and record the number of times each number comes up. You find that the number 3 comes up 23 times. The experimental probability of rolling a 3 on the die is therefore 23/100 or 0.23.  

Example 2: You toss a coin 50 times and record the number of times it lands on heads. You find that it lands on heads 27 times. The experimental probability of flipping heads is therefore 27/50 or 0.54.

Example 3: You draw a card from a deck of 52 cards 200 times and record the number of times you draw a heart. You find that you draw a heart 45 times. The experimental probability of drawing a heart is therefore 45/200 or 0.225.

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Experimental Probability Formula

Here are some common formulas used to calculate probability:

Theoretical Probability Formula: Theoretical probability is the probability of an event based on mathematical calculations. The formula for theoretical probability is:

P(A) = Number of favorable outcomes / Total number of outcomes

where P(A) represents the probability of event A.

Experimental Probability Formula: Experimental probability is the probability of an event based on actual experimentation or observation. The formula for experimental probability is:

P(A) = Number of times event A occurs / Total number of trials or observations

Conditional Probability Formula: Conditional probability is the probability of an event occurring given that another event has already occurred. The formula for conditional probability is:

P(A | B) = P(A and B) / P(B)

where P(A | B) represents the probability of event A given that event B has occurred, P(A and B) represents the probability of both A and B occurring, and P(B) represents the probability of event B occurring.

Multiplication Rule Formula: The multiplication rule is used to calculate the probability of two or more independent events occurring together. The formula for the multiplication rule is:

P(A and B) = P(A) * P(B)

where P(A and B) represents the probability of both A and B occurring, and P(A) and P(B) represent the probabilities of events A and B occurring, respectively.

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Experimental Probability FAQS

What is probability.

Probability is a measure of the likelihood of an event occurring. It is represented as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

What are the types of probability?

There are two main types of probability: theoretical probability and experimental probability. Theoretical probability is based on mathematical calculations, while experimental probability is based on actual experimentation or observation.

What is conditional probability?

Conditional probability is the probability of an event occurring given that another event has already occurred. It is calculated using the formula P(A | B) = P(A and B) / P(B), where P(A | B) represents the probability of event A given that event B has occurred.

What is the difference between independent and dependent events?

Independent events are events in which the occurrence of one event does not affect the probability of the other event occurring. Dependent events are events in which the occurrence of one event affects the probability of the other event occurring.

What is the law of large numbers?

The law of large numbers states that as the number of trials or observations increases, the experimental probability of an event approaches its theoretical probability. This means that with a large enough sample size, the experimental probability becomes more accurate.

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Theoretical and Experimental Probability

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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.

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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 Probability

By the rules of probability, every time you flip a coin, you have an equal chance of getting a head or a tail. Does this mean that for every 10 times you flip the coin, you will always get 5 heads and 5 tails? Or, by the rules of probability, every time you roll a die, you have a 1 in 6 chance of rolling a 4. Does this mean that when you roll the die 60 times, you will definitely roll a 4 ten times? In both of these scenarios, the answer is no. In theoretical probability, you would expect to get heads half the time and tails half the time, and in theoretical probability, you would expect to get each number of pips one-sixth of the time. But in practice, you would be unlikely to get exactly one-half heads or one-sixth 4s. Let''s say that you rolled the die 60 times and rolled a 4 a total of 8 times. In this case, the fraction 8/60 is called the experimental probability. The definition of the experimental probability of an event is the ratio of the number of favorable outcomes to the total number of trials. With a fair die or a fair coin, you know the theoretical probability ahead of time. The more trials you conduct, the closer your experimental probability is likely to get to the theoretical probability. However, experimental probability is more helpful in situations where you don''t or can''t know the probability of the outcome ahead of time.

Examples of experimental probability

Topics related to the experimental probability.

Developing a Probability Distribution from Empirical Data

Probability Distribution

Probability Models

Flashcards covering the Experimental Probability

Statistics Flashcards

Probability Theory Flashcards

Practice tests covering the Experimental Probability

Probability Theory Practice Tests

Common Core: High School - Statistics and Probability Diagnostic Tests

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Tutoring is an excellent way to learn about experimental probability. A tutor can help your student perform actual probability trials to help them learn in a hands-on way how experimental probability works. They can also walk them through the math as many times as needed until your student gains a clear understanding of how experimental probability works. A tutor can answer your student''s questions as soon as they arise so that they learn the information correctly from the beginning. If you''d like to learn more about how tutoring can help your student gain confidence in their understanding of experimental probability, contact Varsity Tutors today and speak with one of our helpful Educational Directors.

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Theoretical vs. Experimental Probability: How do they differ?

Probability is the study of chances and is an important topic in mathematics. There are two types of probability: theoretical and experimental.

So, how to define theoretical and experimental probability? Theoretical probability is calculated using mathematical formulas, while experimental probability is based on results from experiments or surveys. In order words, theoretical probability represents how likely an event is to happen. On the other hand, experimental probability illustrates how frequently an event occurs in an experiment.

Read on to find out the differences between theoretical and experimental probability. If you wonder How to Understand Statistics Easily , I wrote a whole article where I share 9 helpful tips to help you Ace statistics.

Table of Contents

What Is Theoretical Probability?

Theoretical probability is calculated using mathematical formulas. In other words, a theoretical probability is a probability that is determined based on reasoning. It does not require any experiments to be conducted. Theoretical probability can be used to calculate the likelihood of an event occurring before it happens.

Keep in mind that theoretical probability doesn’t involve any experiments or surveys; instead, it relies on known information to calculate the chances of something happening.

For example, if you wanted to calculate the probability of flipping a coin and getting tails, you would use the formula for theoretical probability. You know that there are two possible outcomes—heads or tails—and that each outcome is equally likely, so you would calculate the probability as follows: 1/2, or 50%.

How Do You Calculate Theoretical Probability?

• First, start by counting the number of possible outcomes of the event.
• Second, count the number of desirable (favorable) outcomes of the event.
• Third, divide the number of desirable (favorable) outcomes by the number of possible outcomes.
• Finally, express this probability as a decimal or percentage.

The theoretical probability formula is defined as follows: Theoretical Probability = Number of favorable (desirable) outcomes divided by the Number of possible outcomes.

How Is Theoretical Probability Used in Real Life?

Probability plays a vital role in the day to day life. Here is how theoretical probability is used in real life: 

• Sports and gaming strategies
• Analyzing political strategies.
• Buying or selling insurance
• Determining blood groups 
• Online shopping
• Weather forecast
• Online games

What Is Experimental Probability?

Experimental probability, on the other hand, is based on results from experiments or surveys. It is the ratio of the number of successful trials divided by the total number of trials conducted. Experimental probability can be used to calculate the likelihood of an event occurring after it happens.

For example, if you flipped a coin 20 times and got heads eight times, the experimental probability of obtaining heads would be 8/20, which is the same as 2/5, 0.4, or 40%.

How Do You Calculate Experimental Probability?

The formula for the experimental probability is as follows:  Probability of an Event P(E) = Number of times an event happens divided by the Total Number of trials .

If you are interested in learning how to calculate experimental probability, I encourage you to watch the video below.

How Is Experimental Probability Used in Real Life?

Knowing experimental probability in real life provides powerful insights into probability’s nature. Here are a few examples of how experimental probability is used in real life:

• Rolling dice
• Selecting playing cards from a deck
• Drawing marbles from a hat
• Tossing coins

The main difference between theoretical and experimental probability is that theoretical probability expresses how likely an event is to occur, while experimental probability characterizes how frequently an event occurs in an experiment.

In general, the theoretical probability is more reliable than experimental because it doesn’t rely on a limited sample size; however, experimental probability can still give you a good idea of the chances of something happening.

The reason is that the theoretical probability of an event will invariably be the same, whereas the experimental probability is typically affected by chance; therefore, it can be different for different experiments.

Also, generally, the more trials you carry out, the more times you flip a coin, and the closer the experimental probability is likely to be to its theoretical probability.

Also, note that theoretical probability is calculated using mathematical formulas, while experimental probability is found by conducting experiments.

What to read next:

• Types of Statistics in Mathematics And Their Applications .
• Is Statistics Harder Than Algebra? (Let’s find out!)
• Should You Take Statistics or Calculus in High School?
• Is Statistics Hard in High School? (Yes, here’s why!)

Wrapping Up

Theoretical and experimental probabilities are two ways of calculating the likelihood of an event occurring. Theoretical probability uses mathematical formulas, while experimental probability uses data from experiments. Both types of probability are useful in different situations.

I believe that both theoretical and experimental probabilities are important in mathematics. Theoretical probability uses mathematical formulas to calculate chances, while experimental probability relies on results from experiments or surveys.

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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.