experiment mathematics definition

Something that can be repeated that has a set of possible results.

Examples: • Rolling dice to see what random numbers come up. • Asking your friends to choose a favorite pet from a list

Experiments help us find out information by collecting data in a careful manner.

Experimental Mathematics in Mathematical Practice

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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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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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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.

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.

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• Experiment | Definition & Meaning

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Basics of an Experiment 

Controlled variables, independent variable, dependent variable, types of experiments, experiment|definition & meaning.

An experiment is a series of procedures and results that are carried out to answer a specific issue or problem or to confirm or disprove a theory or body of knowledge about a phenomenon.

Figure 1: Illustration of an Experiment

The scientific method, a methodical approach to learning about the world around you, is founded on the concept of the experiment . Even while some experiments are conducted in labs, you can conduct an experiment at every time and everywhere.

• The key stages of the scientific process are as follows:
• Keenly Observe things.
• Develop a hypothesis .
• Create and carry out an experiment to verify Your hypothesis.
• Analyze the findings of your experiment.
• Based on the analysis of your results, approve or refute your hypothesis.
• Create a new hypothesis , if required, and evaluate it.

Types of Variables in an Experiment

A variable is, to put it simply, everything that can be altered or managed throughout an experiment. Humidity, the length of the study, the structure of an element, the intensity of sunlight, etc. are typical examples of variables. In any study, there are 3 major types of variables :

• Controlled variables (c.v)
• Independent variables (i.v)
• Dependent variables (d.v).

Figure 2: Illustration of Types of Variables

Variables that are maintained constant or unchangeable are known as controlled variables, sometimes known as constant variables. For instance, if you were evaluating the amount of fizz emitted by various sodas, you may regulate the bottle size to ensure that all soda manufacturers were in 12- ounce bottles. If you were conducting an experiment on the effects of spraying plants with various chemicals, you will attempt to keep a similar pressure and perhaps a similar amount when spraying the plants.

The only variable that  you can modify  is the independent variable. It is one factor as you typically try to adjust one element at a time in experiments. As a result, measuring and interpretation of data are made quite simple. For instance, if you’re attempting to establish whether raising the temperature makes it possible to solvate more amount of sugar in the water, the water temperature is the independent variable. This is the factor that you are consciously in control of.

The variable that is  monitored  to determine whether or not your independent variable has an impact is known as the dependent variable. For instance, in the case where you raise the water temperature to observe if it has an impact on the solubility of sugar in it, the weight or volume of sugar (depending on which one you want to calculate) will be the dependent variable.

There are three main types of experiments. Each has its own pros and cons and is carried out according to the nature of the given scenario and desired outcomes. Following are the names of these three types.

Quasi Experiment

Controlled experiment, field experiment.

Figure 3: Illustration of Types of Experiments

Each of these experiments is discussed below along with their strengths and weaknesses.

These are often conducted in a natural environment and involve measuring the impact of one object on another to determine its impact (D.V.). In Quasi-experiments, the research is simply assessing the impact of an event that is already occurring because there is no intentional modification of the variable in this instance; rather, it is changing naturally.

Owing to the unavailability of the researcher, variables occur   naturally , allowing for easy generalization of results to other (real-life) situations, which leads to greater ecological validity.

Absence of control – Quasi-experiments possess poor internal validity since the experimenter cannot always precisely analyze the impact of the independent variable because there is no influence over the environment or other supplementary variables.

Non-repeatable – Because the researcher has no control over the research process, the validity of the findings cannot be verified.

Controlled experiments are also known as lab experiments . Controlled experiments are carried out under carefully monitored conditions, with the researcher purposefully altering one variable (Independent Variable) to determine how it affects another (dependent Variable).

Control – lab studies have a higher level of environmental and other extrinsic variable control, which allows the scientist to precisely examine the impact of the Independent Variable, increasing internal validity.

Replicable – because of the researcher’s greater degree of control, research techniques may be replicated so that the accuracy of the findings can be verified.

Absence of ecological validity — results are difficult to generalize to other (real-life) situations because of the researcher’s participation in modifying and regulating variables, which leads to poor external validity.

A field experiment could be a controlled or a Quasi-experiment. Instead of taking place in a laboratory, it occurs in the actual world. An illustration of a field experiment could be one that involved an organism in its natural environment.

Validity : Because field experiments are carried out in a natural setting and with a certain level of control, they are considered to possess adequate internal and external validity.

Internal validity is believed to be poorer because there is less control than in lab trials, making it more probable that uncontrollable factors would skew results.

An Example of Identifying the Variables in an Experiment

A farmer wants to determine the effect of different amounts of fertilizer on his crop yield. The farmer does not change the amount of water given to the crop for different amounts of fertilizer applied to the field. Determine which of the variables is the controlled variable, independent variable, and independent variable. Also, mention the reasons behind it.

Figure 4: Illustration of the Example

Controlled variable : The amount of water given to the crop is a controlled variable since it is not changed when different amounts of fertilizer are applied.

Independent variable : The amount of fertilizer added to crops is the independent variable. This is because it is the variable which is being manipulated to determine its impact on crop yield.

Dependent variable : Crop yield is the dependent variable in this example. This is because it is the variable on which the impact of the independent variable (Amount of fertilizer) is being monitored .

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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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Experiment: Definitions and Examples

Experiment: Definitions, Formulas, & Examples

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Introduction

Experiments are a fundamental part of mathematics, used to test hypotheses and establish relationships between variables. They are used in many fields, including physics, chemistry, biology, psychology, and economics. In this article, we will explore the basics of experiments in math, including definitions, examples, and a quiz to test your understanding.

Definitions

Before we dive into examples, let’s define some key terms related to experiments in math.

• Experiment: A process used to test a hypothesis or investigate a phenomenon. The process involves manipulating one or more variables and measuring the effect on one or more outcomes.
• Hypothesis: A statement or assumption about a phenomenon that is being tested in an experiment.
• Independent variable: The variable that is being manipulated in an experiment. It is also called the predictor variable or the input variable.
• Dependent variable: The variable that is being measured in an experiment. It is also called the response variable or the output variable.
• Control group: A group in an experiment that does not receive the treatment being tested. It is used as a baseline for comparison with the experimental group.
• Experimental group: A group in an experiment that receives the treatment being tested.
• Randomization: The process of randomly assigning subjects to groups in an experiment. This is done to minimize the effect of confounding variables.

Now that we have defined some key terms, let’s explore some examples of experiments in math.

• A scientist wants to test the effect of caffeine on reaction time. She recruits 100 subjects and randomly assigns them to two groups: one group receives caffeine, and the other group receives a placebo. She then measures their reaction time using a computer-based test.
• A researcher wants to test the effect of a new drug on blood pressure. He recruits 200 subjects and randomly assigns them to two groups: one group receives the new drug, and the other group receives a placebo. He then measures their blood pressure at various time points.
• A teacher wants to test the effect of a new teaching method on student performance. She randomly assigns 50 students to two groups: one group receives the new teaching method, and the other group receives the traditional teaching method. She then measures their performance on a standardized test.
• A psychologist wants to test the effect of music on mood. He recruits 30 subjects and randomly assigns them to two groups: one group listens to classical music, and the other group listens to no music. He then measures their mood using a standardized questionnaire.
• An economist wants to test the effect of a tax cut on consumer spending. He collects data on 100 households and measures their spending before and after the tax cut.
• A physicist wants to test the effect of temperature on the viscosity of a liquid. She heats the liquid to different temperatures and measures its viscosity using a viscometer.
• A biologist wants to test the effect of light on plant growth. She grows plants under different light conditions and measures their height and weight after a specified time.
• A mathematician wants to test the effect of an online tutorial on student understanding of a concept. She randomly assigns 50 students to two groups: one group watches the tutorial, and the other group does not. She then measures their understanding using a standardized test.
• A sociologist wants to test the effect of social media on self-esteem. She recruits 100 subjects and randomly assigns them to two groups: one group uses social media for an hour each day, and the other group does not. She then measures their self-esteem using a standardized questionnaire.
• An engineer wants to test the effect of a new manufacturing process on product quality. He randomly assigns 50 products to two groups: one group is manufactured using the new process, and the other group is manufactured using the traditional process. He then measures their quality using a standardized metric.
• What is the purpose of an experiment in math?

The purpose of an experiment in math is to test a hypothesis or investigate a phenomenon by manipulating one or more variables and measuring the effect on one or more outcomes.

• What is the difference between an independent variable and a dependent variable?

The independent variable is the variable that is being manipulated in an experiment, while the dependent variable is the variable that is being measured in the experiment.

• Why is randomization important in experiments?

Randomization is important in experiments because it minimizes the effect of confounding variables and ensures that the groups being compared are as similar as possible, except for the variable being tested.

• What is a control group?

A control group is a group in an experiment that does not receive the treatment being tested. It is used as a baseline for comparison with the experimental group.

• What is a hypothesis?

A hypothesis is a statement or assumption about a phenomenon that is being tested in an experiment.

• What is the purpose of an experiment in math? A) To test a hypothesis or investigate a phenomenon B) To prove a theory C) To collect data randomly D) None of the above
• What is the difference between an independent variable and a dependent variable? A) The independent variable is the variable being measured, and the dependent variable is the variable being manipulated. B) The independent variable is the variable being manipulated, and the dependent variable is the variable being measured. C) There is no difference between the two. D) Both variables are manipulated.
• Why is randomization important in experiments? A) It ensures that the groups being compared are as similar as possible, except for the variable being tested. B) It ensures that the groups being compared are different in every possible way. C) It has no effect on the outcome of the experiment. D) Both A and B.
• What is a control group? A) A group in an experiment that receives the treatment being tested. B) A group in an experiment that does not receive the treatment being tested. C) A group in an experiment that is randomly assigned to a treatment or no-treatment condition. D) Both A and C.
• What is a hypothesis? A) A statement or assumption about a phenomenon that is being tested in an experiment. B) A group in an experiment that receives the treatment being tested. C) A variable being manipulated in an experiment. D) A variable being measured in an experiment.
• A scientist wants to test the effect of exercise on heart rate. She recruits 50 subjects and randomly assigns them to two groups: one group exercises for 30 minutes, and the other group does not. She then measures their heart rate. What is the independent variable? A) Heart rate B) Group assignment (exercise or no exercise) C) Time D) None of the above
• What is the dependent variable in the experiment described in question 6? A) Heart rate B) Group assignment (exercise or no exercise) C) Time D) None of the above
• What is the purpose of a control group? A) To provide a baseline for comparison with the experimental group. B) To manipulate the independent variable. C) To measure the dependent variable. D) Both B and C.
• A researcher wants to test the effect of a new drug on blood sugar levels. He recruits 100 subjects and randomly assigns them to two groups: one group receives the new drug, and the other group receives a placebo. He then measures their blood sugar levels. What is the experimental group in this experiment?
• A researcher wants to test the effect of a new drug on blood sugar levels. He recruits 100 subjects and randomly assigns them to two groups: one group receives the new drug, and the other group receives a placebo. He then measures their blood sugar levels. What is the experimental group in this experiment? A) The group that receives the new drug B) The group that receives the placebo C) Both groups D) Neither group

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Statistics and probability

Course: statistics and probability   >   unit 6, introduction to experiment design.

• Matched pairs experiment design
• The language of experiments
• Principles of experiment design
• Experiment designs
• Random sampling vs. random assignment (scope of inference)

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Video transcript

Math Definitions - Letter E

• 1» e (Euler's Number)-
• 1» Eccentricity-
• 1» Edge
• 1» Electronic Funds Transfer (EFT)
• 1» Element
• 1» Ellipse
• 1» End Point
• 1» Engineering Notation
• 1» Enlarge
• 1» Equal
• 1» Equality
• 1» Definition Equal Sign
• 1» Equation
• 1» Equation of a Straight Line
• 1» Equiangular
• 1» Equilateral Triangle
• 1» Equidistant
• 1» Equidistant Points
• 1» Equiangular Triangle
• 1» Equinox
• 1» Equity
• 1» Equivalent
• 1» Equivalent Fractions
• 1» Error
• 1» Estimate
• 1» Estimation
• 1» Evaluate
• 1» Even Number
• 1» Event
• 1» Excess
• 1» Expand
• 1» Expanded Notation
• 1» Experiment
• 1» Exponent
• 1» Expression
• 1» Exterior Angle
• 1» Extraneous Solution
• 1» Extrema

Definition of Experiment

An experiment is a way of collecting data.

For example, if you want to find out how likely it is for a spinner to land on "yellow", a useful experiment would be to spin the spinner one hundred times, note down which colour it lands on each time, and calculate the fraction of these that are yellow.

Another experiment might involve asking your friends to tell you their favourite computer game and writing down the results.

Description

The aim of this dictionary is to provide definitions to common mathematical terms. Students learn a new math skill every week at school, sometimes just before they start a new skill, if they want to look at what a specific term means, this is where this dictionary will become handy and a go-to guide for a student.

Year 1 to Year 12 students

Learning Objectives

Learn common math terms starting with letter E

Author: Subject Coach Added on: 6th Feb 2018

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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.

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Definition of an "Experiment" in Probability

One can define the fundamental concepts of probability theory (such as a probability measure, random variable, etc) in a purely axiomatic manner. However, when we teach probability, we start off with the notion of an "experiment", a concept it seems to me which is something akin to pornography: difficult to define, but you tend to know it when you see it.

So I am curious if there is a general definition of an experiment (or if it something really best regarded more as an explanatory construct). To try to define an experiment as a type of function seems difficult to me b/c it would require the notion of a "random function" of some type.

Thanks, Jack

• probability

4 Answers 4

I like the way it is defined in Mathematical Statistics By Wiebe R. Pestman:

"A probability experiment is an experiment which, when repeated under the same conditions, does not necessarily give the same results"

This is useful as well.

• $\begingroup$ My initial reaction is to say that I'm gobsmacked. I'll have to read the article though to understand the context (and his meaning) better. Is this what most statisticians take to be the definition, do you think? Thanks $\endgroup$ –  Jack Zega Commented Feb 20, 2013 at 23:47
• $\begingroup$ @JackZega By no means I'm a statistician. The above tries to convey difference between deterministic experiment versus probability experiments. For example, if you try to measure voltage $V$ through a certain conductor with resistance $R$, you will always end up with the value $IR$ no matter what (this is a deterministic experiment ). One advice, the deeper you go the more hair you lose :) $\endgroup$ –  jay-sun Commented Feb 20, 2013 at 23:55
• $\begingroup$ I think this definition, taken out of its context, is quite dangerous. A probabilistic experiment must be infinitely repeatable. By infinitely repeatable we mean that each time you repeat the experiment, the sample space $\Omega$ (and so everything that follows from it) is constant. So actually, an experiment is a procedure that, when repeated keeps the sample space constant. Therefore the possible results of an experiments are always the same, but the actual result of a repetition of an experiment, could be different, if the experiment is a random experiment and not a deterministic one. $\endgroup$ –  Euler_Salter Commented Aug 18, 2018 at 14:03

One would think that it would be the other way round - everyone understands what it means to roll a die, but the notion of a random variable is far less trivial.

To define an experiment, first define a "generator" - any physical or algorithmic method for producing $N$ numbers, such that $N$ tends to infinity, the numbers produced are distributed according to random variable $X$.

The production of any individual number using a generator is an experiment.

“Experiment is a systematic way of varying all the factors of interest and observing impact of these all factors on the desired output.”

While reading Grimmett & Welsh book, I found that an experiment is

Any procedure whose consequences are not predetermined.

This is quite a restrictive definition in my opinion, because it excludes "deterministic" experiments where we can determine the final result 100% precision (because there is actually only one possible result). So to me, it the following definition given by Wikipedia seems more precise:

Any procedure that is infinitely repeatable and whose outcomes are well-defined

This seems fine, but it doesn't seem tremendously mathematical. So my guess is that the definition above is correct, and that is the definition used to define a sample space, outcomes, events, event space, probability measure, and so on. However, once we've defined all those mathematical structures, we can go back and say: actually, an experiment can easily be represented by a probability space $(\Omega, \Sigma, \mathbb{P})$. This could be seen as a circular definition, but if you use the word "represented" instead of "defined" then you should be fine.

I just wrote an article (which I will extend soon, it's still under construction) in my website SimpleAI

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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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