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Experimental Probability

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Experimental Probability vs. Theoretical Probability

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

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

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

Theoretical probability for Event A can be calculated as follows:

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

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

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

Experimental Probability: Examples

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

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

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

             $= \frac{4}{10}$

             $= \frac{2}{5}$

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

The experimental probability of rolling a 2 

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

$= \frac{5}{20}$

$= \frac{1}{4}$

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

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

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

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

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

               $= \frac{10}{25}$

               $= \frac{2}{5}$

               $= 0.4$

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

Solution: 

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

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

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

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

Number of times 3 showed $= 7$

Number of tosses $= 30$

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

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

Solution:  

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

So, the number of trials $= 20$

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

$= \frac{4}{5}$

$= 0.8$ or $80%$

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

Number of white bikes $= 100$ 

Total number of bikes $= 500$

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

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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

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.

In probability and statistics, an experiment typically refers to a study in which the experimenter is trying to determine whether there is a relationship between two or more variables. In an experiment, the subjects are randomly assigned to either a treatment group or a control group (there can be more than one of either group).

Generally, the control group in an experiment receives a placebo (substance that has no effect) or no treatment at all. The treatment group receives the experimental treatment. The goal of the experiment is to determine whether or not the treatment has the desired/any effect that differs from the control group to a degree that the difference can be attributed to the treatment rather than to random chance or variability. Well-designed experiments can yield informative and unambiguous conclusions about cause and effect relationships.

As an example, if a scientist wants to test whether a new medication they developed has any effect, they would select subjects from a common population and randomly assign them to either a treatment group or a control group. They would then administer the treatment to the treatment group, and either a placebo or no treatment to the control group, and study the effects of each using statistical measures to determine whether the medication had any effect beyond chance or variability.

Note that an experiment does not necessarily need to have a physical treatment. The term "treatment" is used fairly loosely. Another experiment could look at the effects of getting advice from a college counselor on admission rates compared to not getting advice from a college counselor. In this case, the "treatment" would be getting advice from a college counselor. The control group would get no advice from a college counselor.

Importance of experimental design

Like survey methodology , experimental design is essential to the validity of the results of the experiment. A poorly designed experiment can result in false or incorrect conclusions. Proper statistical experiment design generally involves the following:

• Identification of the explanatory variable, also referred to as the independent variable . The explanatory variable is the "treatment," or the thing that causes the change, and can be anything that causes a change in the response variable.
• Identification of the response variable, also referred to as the dependent variable . It is the variable that may be affected by the explanatory/independent variable.
• Defining the population of interest and taking a random sample from the population. Generally the larger the random sample, the less potential for sample error, since the larger sample will likely be more representative of the population.
• Random assignment of the subjects in the sample to either the treatment group or the control group.
• Administration of the treatment to the treatment group, and placebo (or nothing) to the control group), possibly using a blind experiment (the subject doesn't know whether they are receiving the treatment or the placebo) or double blind experiment (neither experimenter nor subject knows which treatment they are getting).
• Measurement of the response over a chosen period of time.
• Statistical analysis of the supposed response to determine whether there is an actual response, or the response can be attributed to chance, to determine whether there is a causal relationship between the treatment and the response.
• Replication of the experiment by peers, assuming there is a causal relationship between the treatment and the response.

Experiments vs surveys

Experiments and surveys are both techniques used as part of inferential statistics . A survey involves the use of a random sample of the population, rather than the whole, with the goal that all subjects in the population have an equal chance of being selected. The random sample of the population is then used to draw conclusions or make inferences about the population as a whole.

In contrast, an experiment typically involves the use of random assignment such that all subjects have an equal chance of being assigned to the groups (treatment and control) in the study, which minimizes potential biases, as well as allows the experimenters to evaluate the role of variability in the experiment. This in turn allows them to determine whether any observed differences between the groups merit further study or not based on whether or not the differences can be attributed to variability or chance.

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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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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 Mathematics

Experimental mathematics is a type of mathematical investigation in which computation is used to investigate mathematical structures and identify their fundamental properties and patterns. As in experimental science, experimental mathematics can be used to make mathematical predictions which can then be verified or falsified on the bases of additional computational experiments.

Borwein and Bailey (2003, pp. 2-3) use the term "experimental mathematics" to mean the methodology of doing mathematics that includes the use of computation for:

1. Gaining insight and intuition.

2. Discovering new patterns and relationships.

3. Using graphical displays to suggest underlying mathematical principles.

4. Testing and especially falsifying conjectures.

5. Exploring a possible result to see if it is worth formal proof.

6. Suggesting approaches for a formal proof.

7. Replacing lengthy hand derivations with computer-based derivations.

8. Confirming analytically derived results.

Examples of tools of experimental mathematics include computer algebra , symbolic algebra , Gröbner basis , integer relation algorithms (such as the LLL algorithm and PSLQ algorithm ), arbitrary precision numerical evaluations, computer visualization, cellular automata and related structures, and databases of mathematical structures such as the Online Encyclopedia of Integer Sequences ( http://www.research.att.com/~njas/sequences ) by Neil Sloane, The Wolfram Functions Site ( http://functions.wolfram.com ) by Michael Trott and Oleg Marichev, and MathWorld ( http://mathworld.wolfram.com ) by Eric Weisstein.

Explore with Wolfram|Alpha

More things to try:

• expand (x^2 + 1)(x^2 - 1)(x+1)^3
• multibrot set, d=6

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Cite this as:.

Weisstein, Eric W. "Experimental Mathematics." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/ExperimentalMathematics.html

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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 Mathematics in Mathematical Practice

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• First Online: 04 November 2023
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• Jessica Carter 2  

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This chapter presents an overview of the contributions to the section on Experimental Mathematics by focusing in particular on how they characterize the phenomenon of “experimental mathematics” and its origins. The second part presents two case studies illustrating how experimental mathematics is understood in contemporary analysis. The third section offers a systematic presentation of the contributions to the section.

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Taken from the webpage of Experimental Mathematics , see https://www.tandfonline.com/action/journalInformation?show=aimsScope&journalCode=uexm20

The quote is from Peirce’s Collected Papers 1931 –1960, volume III, paragraph 363. I refer to (Marietti 2010 ) and (Carter 2020 ) for explanations of Peirce’s diagrammatic reasoning.

The fact that the result is most likely undecidable contradicts Baker’s ( 2008 ) statement that mathematics need not rely on inductive methods since it is always possible that a formal, or deductive, proof can be found of a given proposition (p. 337).

I thank W. Szymanski for conversations about these cases.

There are further technical restrictions imposed on the graphs that are not relevant for this brief presentation.

The concept of an amenable group has been introduced by von Neumann in connection with his work on the Banach-Tarski paradox.

The group commutator [ g , h ] is given by the expression [ g ,  h ] =  ghg −1 h −1 .

A previous well-known method to divide a line segment in equal parts depends on the stronger assumption that it is possible to draw parallel lines.

Avery J, Johansen R, Szymanski W (2018) Visualizing automorphisms of graph algebras. Proc Edinb Math Soc 61(1):215–249

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Avigad J (2018) Opinion: the mechanization of mathematics. Not Am Math Soc 65(6):681–690

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Baker A (2008) Experimental mathematics. Erkenntnis 68:331–344

Borwein JM (2012) Exploratory experimentation: digitally-assisted discovery and proof. In: Hanna G, de Villiers M (eds) Proof and proving in mathematics education, New ICMI study series 15. Springer, New York, pp 69–96. https://doi.org/10.1007/978-94-007-2129-6_4

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Carter J (2020) Logic of relations and diagrammatic reasoning: structuralist elements in the work of Charles Sanders Peirce. In: The prehistory of mathematical structuralism. Oxford University Press, New York, pp 241–272

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Haagerup S, Haagerup U, Ramirez-Solano M (2021) Computational explorations of the Thompson Group T for the amenability problem of F. Exp Math 30(1):105–126

Hersh R (1991) Mathematics has a front and a back. Synthese (Dordrecht) 88(2):127–133

Marietti S (2010) Observing signs. In: Moore ME (ed) New essays on Peirce’s mathematical philosophy. Open Court, Chicago/La Salle, pp 147–167

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Peirce CS (1931–1960) Collected Papers of Charles Sanders Peirce, Vol I–IV, Hartshorne C, Weiss P (eds), The Belknap Press of Harvard University Press, Cambridge

Tymoczko T (1979) The four-color problem and its philosophical significance. J Philos 76(2):57–83

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Van Bendegem J-P (1998) What, if anything, is an experiment in mathematics? In: Anapolitanos D, Baltas A, Tsinorema S (eds) Philosophy and the many faces of science. Rowman & Littlefiel, London, pp 172–182

Zeilberger D (1994) Theorems for a price: tomorrow’s semi-rigorous mathematical culture. Math Intell 16(4):11–18

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Jessica Carter

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Carter, J. (2023). Experimental Mathematics in Mathematical Practice. In: Sriraman, B. (eds) Handbook of the History and Philosophy of Mathematical Practice. Springer, Cham. https://doi.org/10.1007/978-3-030-19071-2_121-1

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Probability

How likely something is to happen.

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

Tossing a Coin

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

Heads (H) or Tails (T)

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

Throwing Dice

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

The probability of any one of them is 1 6

In general:

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

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

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

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

So the probability = 1 6

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

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

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

So the probability = 4 5 = 0.8

Probability Line

We can show probability on a Probability Line :

Probability is always between 0 and 1

Probability is Just a Guide

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

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

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

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

Learn more at Probability Index .

Some words have special meaning in Probability:

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

Example: Throwing dice

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

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

Outcome: A possible result.

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

Trial: A single performance of an experiment.

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

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

Sample Space: all the possible outcomes of an experiment.

Example: choosing a card from a deck

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

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

The Sample Space is made up of Sample Points:

Sample Point: just one of the possible outcomes

Example: Deck of Cards

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

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

There are 6 different sample points in that sample space.

Event: one or more outcomes of an experiment

Example Events:

An event can be just one outcome:

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

An event can include more than one outcome:

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

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

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

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

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

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

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

These are Alex's Results:

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

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Different mathematical solving methods can affect how information is memorized

by University of Geneva

The way we memorize information—a mathematical problem statement, for example—reveals the way we process it. A team from the University of Geneva (UNIGE), in collaboration with CY Cergy Paris University (CYU) and Bourgogne University (uB), has shown how different solving methods can alter the way information is memorized and even create false memories.

By identifying learners' unconscious deductions, this study opens up new perspectives for mathematics teaching. These results are published in the Journal of Experimental Psychology: Learning, Memory, and Cognition .

Remembering information goes through several stages: perception, encoding—the way it is processed to become an easily accessible memory trace—and retrieval (or reactivation). At each stage, errors can occur, sometimes leading to the formation of false memories .

Scientists from the UNIGE, CYU and Bourgogne University set out to determine whether solving arithmetic problems could generate such memories and whether they could be influenced by the nature of the problems.

Unconscious deductions create false memories

When solving a mathematical problem , it is possible to call upon either the ordinal property of numbers, i.e., the fact that they are ordered, or their cardinal property, i.e., the fact that they designate specific quantities. This can lead to different solving strategies and, when memorized, to different encoding.

In concrete terms, the representation of a problem involving the calculation of durations or differences in heights (ordinal problem) can sometimes allow unconscious deductions to be made, leading to a more direct solution. This is in contrast to the representation of a problem involving the calculation of weights or prices (cardinal problem), which can lead to additional steps in the reasoning, such as the intermediate calculation of subsets.

The scientists therefore hypothesized that, as a result of spontaneous deductions, participants would unconsciously modify their memories of ordinal problem statements, but not those of cardinal problems.

To test this, a total of 67 adults were asked to solve arithmetic problems of both types, and then to recall the wording in order to test their memories. The scientists found that in the majority of cases (83%), the statements were correctly recalled for cardinal problems.

In contrast, the results were different when the participants had to remember the wording of ordinal problems, such as: "Sophie's journey takes 8 hours. Her journey takes place during the day. When she arrives, the clock reads 11. Fred leaves at the same time as Sophie. Fred's journey is 2 hours shorter than Sophie's. What time does the clock show when Fred arrives?"

In more than half the cases, information deduced by the participants when solving these problems was added unintentionally to the statement. In the case of the problem mentioned above, for example, they could be convinced—wrongly—that they had read: "Fred arrived 2 hours before Sophie" (an inference made because Fred and Sophie left at the same time, but Fred's journey took 2 hours less, which is factually true but constitutes an alteration to what the statement indicated).

"We have shown that when solving specific problems, participants have the illusion of having read sentences that were never actually presented in the statements, but were linked to unconscious deductions made when reading the statements. They become confused in their minds with the sentences they actually read," explains Hippolyte Gros, former post-doctoral fellow at UNIGE's Faculty of Psychology and Educational Sciences, lecturer at CYU, and first author of the study.

Invoking memories to understand reasoning

In addition, the experiments showed that the participants with the false memories were only those who had discovered the shortest strategy, thus revealing their unconscious reasoning that had enabled them to find this resolution shortcut. On the other hand, the others, who had operated in more stages, were unable to "enrich" their memory because they had not carried out the corresponding reasoning.

"This work can have applications for learning mathematics. By asking students to recall statements, we can identify their mental representations and therefore the reasoning they used when solving the problem, based on the presence or absence of false memories in their restitution," explains Emmanuel Sander, full professor at the UNIGE's Faculty of Psychology and Educational Sciences, who directed this research.

It is difficult to access mental constructs directly. Doing so indirectly, by analyzing memorization processes, could lead to a better understanding of the difficulties encountered by students in solving problems, and provide avenues for intervention in the classroom.

Provided by University of Geneva

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3 Math Tasks You Already Use That Can Foster Collaboration

Elementary teachers can turn common activities into opportunities for students to work and learn together.

Picture a “traditional” math class and then a “21st-century” math class. One of the most obvious differences is probably how students are interacting: No longer silently sitting in rows of desks, students are talking to each other. They are collaborating. 

Collaborative learning promotes higher-level reasoning, self-esteem, and positive relationships at school . To unlock the benefits of collaborative learning for young mathematicians, teachers can plan accessible tasks. When students don’t have to devote as much “brain space” to understanding the directions for a task, they can focus instead on collaborating on the math itself. This supports all students, including multilingual students and students with disabilities.

Start with tasks that are engaging, but not content-heavy , so that you can prioritize laying the groundwork for collaboration. Even a task as simple as taking turns drawing a picture can be productive for discussing expectations around sharing work and communicating. From there, you can use the warm-ups, word problems, and manipulatives that you and your students are already familiar with to generate accessible and meaningful collaborative tasks.

1. Collaborate Using Your Warm-Ups

Routines like warm-ups, number sense routines, or number talks invite students to reason and discuss, typically in a whole group setting. However, these can be done collaboratively as well. 

Selecting warm-ups for collaboration: Look for routines that have many possible “right” answers, so that all group members will be able to contribute meaningfully. This doesn’t have to be multiple numerical answers; it could also be many ways to get to a solution—for example, different ways to see a total in a dot talk . When you do this routine as a whole group, you may find yourself saying to students, “Wow, I hadn’t thought of that!” Those activities lend themselves well to collaboration.

Preparing warm-ups for collaboration: In a whole group discussion, you use your learning objectives to steer the conversation as it happens. For example, if we were discussing “Ways to Make 1/8” as a whole class and students were only talking about shapes, I might ask the class to think about a number line. If this discussion were to happen in collaborative groups, that steering needs to be built into the task. One way to do this is with a checklist. For example, when my third graders did “Ways to Make 1/8” as a collaborative task, each group got this checklist:

• 2 ways to make 1/8 of a square
• 2 ways to make 1/8 of a rectangle
• 1 way to make 1/8 of a circle
• 1 way to make 1/8 on a number line
• 2 ways to make 1/8 of numbers (example: 1/8 of 16 is 2)

The checklist makes it obvious to students how they should be contributing, and even though I might not be part of every conversation, I know that key points are surfacing throughout the room.

2. Collaborate Using Your Word Problems

Whether writing word problems yourself or using the ones in a curriculum, word problems are a feature of every math class.

Selecting word problems for collaboration: Make sure the context of the chosen word problem is accessible, or plan for making the context accessible. You might use images or videos to make sure students understand the vocabulary and can make connections to background knowledge. If the word problems in your curriculum have instructions like “Use an array model to solve,” leave those parts out. Groups should solve in ways that make sense to them, and diversity in strategies both within groups and between groups will lead to important mathematical discussions.

Preparing word problems for collaboration: You can turn one word problem into an extended collaborative task by planning a series of increasingly challenging number sets to fit into the story. For example, “Allison filled up 8 baskets of peaches, with 10 peaches in each basket. How many peaches is that?” followed by 18 baskets of 10, then 18 baskets of 25. If groups are working at vertical surfaces like whiteboards, they can look around the room after finishing one number set to see what number set to work on next, as described by Peter Liljedahl in Building Thinking Classrooms . This promotes knowledge mobility and frees the teacher to discuss with groups as needed. 

3. Collaborate Using Your Manipulatives

Counting collections is a rich collaborative task for students across elementary grades. You might be surprised by how excited students are to count even the most familiar math manipulatives. 

Preparing counting collections: Students will need “right-sized” collections, tools for organizing, and space to record. Don’t be afraid to offer big collections. Collections of, say, 45 for kindergartners or 845 for fourth graders can be meaningful opportunities for developing place value understanding. Tools like cups, plates, paper trays from the cafeteria, and your number lines or hundred charts should be available for students to take up as they need. Finally, think about how you want students to record their collections: on paper , on whiteboards, and/or digitally on a platform like Seesaw.

To grow your collection of collections beyond your classroom manipulatives and supplies, invite students to bring in collections from home, or round up unwanted items around your school. I have many beautiful collections of transparent manipulatives left over from the overhead projector days.

If At First You Don’t Succeed…

Collaboration is tricky. Even with extensive planning and coaching , students and groups will struggle at times. But if collaboration is hard for your class , that means they need to practice it more, not less. More collaboration doesn’t have to mean planning more tasks from scratch; you can and should use the resources you already have and turn those into revisit-able routines. Having familiar routines for collaboration means that students can dive straight into the learning, while continuing to develop their identities as good group-mates who know how to support each other and work together.