experiment in math

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 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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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 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%

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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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• 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 .

All images/mathematical drawings were created with GeoGebra.

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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

Applied math

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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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Mathematical Experiments—An Ideal First Step into Mathematics

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Since the foundation of the Mathematikum, Germany, in 2002 and Il Giardino di Archimede, Florence, Italy, in 2004 there have been many activities around the world to present mathematical experiments in exhibitions and museums. Although these activities are all very successful with respect to their number of visitors, the question arises what is their impact for “learning” mathematics in a broad sense. This question is discussed in the paper. We present a few experiments from the Mathematikum and shall then discuss the questions, as to whether these are experiments and whether they show mathematics. The conclusion will be that experiments provide an optimal first step into mathematics. This means in particular that they do not offer the whole depth of mathematical reasoning, but let the visitors experience real mathematics, insofar as they provide insight by thinking.

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Mathematics, Aesthetics, and the Arts

• Mathematical experiments
• Science centers
• Learning by experience
• Mathematics in leisure time

In the last years, quite a few mathematical exhibitions have been developed and mathematical museums (“science centers”) have been opened. In these, mathematics is typically not presented in the traditional way using the mathematical language. On the contrary: visitors find “exhibits”, in which they may see or explore mathematics. In other words, visitors are challenged to perform “mathematical experiments”. In addition, also several books with easy-to-perform experiments have been published, which aim at teachers, students or the general public.

In this article we look at mathematical experiments, and investigate their potential for formal and informal learning of mathematics. The basic reason for the success of science centers in general is expressed in the slogan “hands-on, minds-on, hearts-on”. In other words, in performing the experiments, visitors get experience. This experience leads to understanding, and understanding gives pleasure.

2.1 Mathematical Experiments and Science Centers

Probably the first man-made experiments are due to the time of Galilei (for instance experiments with pendula). In mathematics, models and instruments became important in the 19th century. The book of Dyck ( 1892 ) shows an impressive collection of mathematical models, apparatuses and instruments.

Some mathematical experiments have been known for a long time, mostly under the name of “mathematical games”. Famous games are for instance Hamiltons’s Icosian Game (1857), the “Tower of Hanoi” (Lucas 1883), and the Soma cube (Hein 1934).

The first initiative to collect and develop mathematical experiments as such was undertaken by the Italian professors Franco Conti and Enrico Giusti, who very successfully developed and organized the exhibition “Oltre iI compasso—the mathematics of curves”, which was first shown in 1992. Since 2004 it has been enlarged to form the “Giardino di Archimede” in Florence. Nearly at the same time, the first step towards the Mathematikum was taken: in 1994 the first German exhibition under the name “hands-on mathematics” (“Mathematik zum Anfassen”) was shown in Giessen, Germany. This exhibition was a work of a group of students, who organized this exhibition as a follow-up of a mathematical seminar. Mathematikum, the world’s first mathematical science center, was opened in 2002. Since then, quite a few institutions of different size followed these ideas, for instance “Adventure Land Mathematics” in Dresden, “MoMath” in New York, and “Maison des Maths” in Mons, Belgium.

The idea of all these institutions is basically that the combination “interactive exhibits and visitors” works. It is fascinating to observe that in all science centers visitors start working, without a guide, without a teacher, even without reading the label, and have lots of fun. In most science centers, certainly in all mathematical science centers, the responsible people take science serious. “Fun” should not arise from strange colors, noise, fog and so on, but from insight into the phenomena. Looking at the visitors, we see experience, understanding and pleasure. In the science center-terminology: hands-on, minds-on, hearts-on.

2.2 Mathematikum Giessen

The Mathematikum in Giessen, Germany (near Frankfurt) is a mathematical science centre founded in 2002. It aims to make mathematics accessible to as many people as possible, in particular to young people. On its 1200 m 2 exhibition area it shows about 180 interactive exhibits. From the very beginning, it was a great success. Between 120,000 and 150,000 people visit the Mathematikum each year. About 40% are group visitors, mainly school classes, 60% are private visitors, mainly families.

Visitors like the Mathematikum. In particular they like the way mathematics is presented. They are entertained by performing the experiments and trying to understand what they have experienced. The Mathematikum is a house full of communication. When one listens to what people are talking about, one notices that it is always about the exhibits.

The permanent exhibition of Mathematikum is complemented by several other formats, which address different target groups.

Temporary exhibitions on special topics, such as randomness, calculating devices, mathematics in everyday life, mathematical games, etc.

Popular lectures on special topics such as cryptography, astronomy, etc.

Lectures for children on topics as, for instance, mathematics and—the bicycle, the bees, the heaven, the kitchen, the Christmas tree, and so on.

2.3 Some Experiments

The experiments in Mathematikum cover many mathematical disciplines, such as geometry (shapes and patterns), arithmetic (numbers and calculating), calculus (functions), probability (randomness and statistics), algorithms, and history of mathematics. No mathematical discipline is generally excluded.

We shorty describe some exhibits; more can be found in Beutelspacher ( 2015 ).

Conway’s cube

Figure 2.1 shows an invention of John H. Conway. It is a puzzle consisting of three small cubes of side length 1 and six 2 × 2 × 1-cuboids, which should be assembled to form a cube. One first calculates how big the cube will be. Even with this knowledge, most people struggle—until they get the idea where to locate the small cubes in the big cube.

In Fig. 2.2 we see seven lamps in a circle. To each lamp a switch is attached. When trying the switches, one notes that each switch affects three lamps, precisely the lamp is attached to the switch and the lamps on the right hand side and on the left hand side of the switch. When activating a switch, the status of these three lamps changes: those which have been off, are on now, and those which have been on, are off now.

The task, which is already included in the title of the experiment, is to put all lamps on.

Many people start by randomly pressing the switches. Also in this way, we arrive at situations which are promising. For instance, if four lamps in a row are on, then it is easy to switch on the remaining lamps. Also, if only one lamp is on, one has a promising situation. By pushing one switch one gets four enlightened lamps in a row and one can proceed to finish as above.

Tetrahedron in the cube

The experiment shown in Fig. 2.3 consists of two parts.This experiment consists of two parts. One part is a cube made of glass with its upper face removed. The other part is a rather big tetrahedron which is supposed to be put inside of the cube. Most likely, first attempts will fail. Describing failed attempts, one gets an idea of how to succeed. If one holds the tetrahedron so that one vertex points downwards, it won’t work. Also, if one vertex points upwards (and its face downwards), it will not work. Now, one could think of trying to let an edge point downwards. In fact, putting edge on a diagonal of the cube’s upper square the tetrahedron automatically slides inside.

All triangles are equal

The experiment shown in Fig. 2.4 provides a challenging task. There is a poster showing a pattern of equilateral triangles. Following the task, one has to hold a framed irregular triangle in-between the lamp and the poster. Of course, we see a shadow. Moreover, the shadow is an irregular triangle. The task is now to put the triangle in a position so that its shadow perfectly fits onto one of the smaller equilateral triangles. For this, one has to move the triangle; back and forth, rotating in all possible ways. Eventually, the perfect shadow is found.

The smarties

In the experiment shown in Fig. 2.5 the vistor is confronted with a poster, where one sees an incredible number of smarties, far too many to be counted. If you want to know how many smarties there are, you have to rely on estimation strategies. Estimations nevertheless are not blindly guessing a number but using the method of a random sample. Next to the picture, you find a square frame. Holding the frame onto the picture, it is easy to count the smarties within the frame. Now, you only have to know how often the frame fits into the picture. You find this number for example by how many times the frame fits into the upper side of the picture and how many times it fits into the vertical side.

Pythagoras puzzle

Figure  2.6 shows an experiment related to Pythagoras’ theorem. In front, there is a triangle the longer side of which is blue and the shorter sides are red and yellow. On either side, there is a square which can be filled with coloured plates. The yellow square can be filled with 3 × 3 yellow plates, the red square can be filled with 4 × 4 red plates so that all plates are used. By turning these 9 + 16 plates, the 25 blue plates perfectly fit into the 5 × 5 square above the triangle’s blue side.

This experiment illustrates the Pythagorean theorem which states that in a rectangular triangle, the size of the legs’ squares (a 2  + b 2 ) equals the size of the hypotenuse’s square (c 2 ). In short, a 2  + b 2  = c 2 .

Two in a row

In the experiment shown in Fig. 2.7 six wheels invite us to turn them. Each wheel has colorful pieces on it, which vary in form and color. Four different shapes (triangle, square, star, and circle) occur in four different colors, so that we have in total 16 symbols. Each wheel is adorned with these 16 symbols in some random order.

Now we let the wheels rotate. The wheels come to a standstill at random positions. The question is, whether “by accident” two equal symbols (shape and color identical) are at the same line.

Naively, we would conjecture that this will be a rare event, since it is no problem at all to put the six wheels in a position where no equal symbols are at the line. But when performing the experiment, we often see the bewildering situation that two equal symbols are in the same row.

2.4 Books and Easy-to-Built Experiments

In recent years, quite a few books have appeared which contain experiments with cheap material. Many of them are based on paper folding, assembling objects with sticks, and so on. Classical books on this subject are van Delft and Botermans ( 1978 ), and, on a higher level, Cundy and Rollett ( 1952 ), and Wenninger ( 1974 ).

Most of these books aim at the leisure market (for instance Beutelspacher and Wagner 2008 ), but some are explicitly meant for teachers (e.g. Schmitt-Hartmann and Herget 2013 ).

The experiments in science centers and the models which can be built using these books share several properties.

Everybody can perform it. The experiments are deliberately simply to perform. In a strong way it is “mathematics for everybody”.

People like it. One reason why people like the experiments is their success. Each experiment has the possibility of a positive ending, and “all’s well that ends well”. What is more: the success is undoubtable. When I have composed the pyramid, it stands there and nobody can question it.

On the other hand, from a mathematician’s point of view, people often stop at an early stage and are satisfied with a superficial effect.

2.5 Two Critical Questions

2.5.1 are these experiments at all.

One of the main features of mathematics is that the truth of an assertion is obtained by a proof, that is by purely logical arguments, and not, for instance, by experiments. This distinguishes mathematics from sciences such as physics or chemistry, where experiments are used to verify a theory or to falsify a wrong hypothesis.

Also, mathematical experiments are not used to simply illustrate a definition or a theorem.

The role of a mathematical experiment is quite different. Its basic property is to stimulate thinking. In science centers, experiments do not come second (after a theory), but experiments come first. They provide a strong impulse. Basically, a person working with a mathematical experiment is challenged by a mathematical problem. As in research, one has to get the right conception, the right idea of what’s going on. And sometimes, after a while of thinking, and sometimes with luck, one finds the solution.

A big advantage of such experiments is the fact that the solution is beyond any doubt, because it is materialized: the cube is there, the bridge is stable, the pattern is correct.

To put it short, a mathematical experiment works “bottom-up”: starting from experience, leading to insight. It is an impulse. If the experiment is good, this impulse is so strong that it enables the visitor to ask the right questions, to get the right conceptions and, finally to get by an “Aha-moment” the right insight.

2.5.2 Is This at All Mathematics?

Certainly, it does not look like mathematics, in particular not like school mathematics. In fact, in Mathematikum we explicitly stated at the beginning that we want to make a place that doesn’t look like school. Mathematical experiments do not show the mathematical language: no point is called “P”, no variable is called x, in fact, there are no formulas. Also, no definitions, no theorems, no proofs.

On the other hand, an important part of mathematical activity is clearly present, namely problems. And, if visitors solve the problems, they activate mathematics-related competences, such as arguing, and communicating.

Mathematical experiments have two main target groups. (a) School classes, (b) private visitors.

When a school class visits Mathematikum, the students may deal with experiments closely related to the topics in math education. For instance, they may look at experiments dealing with the theorem of Pythagoras, or with number systems or with randomness. The teacher then can talk with the students the next day in school about their experiences and insights.

Private visitors, in particular families, behave quite differently. First of all, they have no idea, whether an exhibit represents important mathematics or mathematics at all. They do not care whether the formal mathematics behind it is difficult or easy.

For all visitors it is true that when they start to deal with an experiment, they have a chance to perform a first step into mathematics. The most important aspect is that they think. In fact, they automatically start thinking, for instance asking questions and making conjectures. They try out ideas to solve the problem and eventually they experience the Aha!-moment, in which the whole situation becomes clear.

In addition, when trying to solve a mathematical experiment, the visitors concentrate on important mathematical notions such as edge, angle, it fits, etc. and also they get acquainted with important mathematical concepts, such as patterns, correspondence, infinity, etc.

Finally, they meet not only mathematics taught in school but many aspects which go far beyond school, for instance the travelling salesman problem, minimal surfaces, prime numbers, conic sections, etc.

To sum up, working with mathematical experiments is a first step into mathematics. This statement has two sides.

Firstly, it is a step into mathematics. In fact, the problems posed by the experiments can only be solved by thinking, by carefully observing, by looking for the right idea.

On the other hand, dealing with experiments provides only a first step into mathematics. Many more steps could follow. In particular, in this context, there is no formal description of the mathematical phenomena.

In other words, mathematical experiments offer extremely good possibilities to “do” mathematics, but have also clear limitations: they stimulate enthusiasm and true motivation, but also they neither give formal arguments nor can replace a course in a mathematical subject.

Working with mathematical experiments goes far beyond “learning mathematics”. It empowers people: When visitors see that they have achieved something by thinking by themselves, they become more self-confident.

2.6 Effects and Impact on the Visitors

The main effect of all science centers is experience . Visitors experience real phenomena. This is also what visitors like. It is not a virtual experience, which we have by working with computer programs. When we feel real physical objects and work with them, it is clear that we cannot be cheated.

Mathematical experiments stimulate thinking . One has to consider several possibilities, one has to develop the right idea for a solution and one verifies whether a solution is correct.

The unquestionable experience of many years of Mathematikum is that dealing with mathematical experiments makes the visitors happy. They become happy because they have understood something, which is very satisfying (see also Beutelspacher 2016 ).

The fact that experiments activate people’s brain can be seen—or heard—by the noise in the exhibition. Sometimes it is really loud. But in fact, it is communication . People talk to each other, ask questions, give advice—and enjoy the common solution.

A final point: if mathematics is interesting, then it is also interesting outside school. In mathematical science centers as the Mathematikum, mathematics is part of the visitor’s leisure time. Adult people and whole families spend hours to experience the power of mathematics. Thus, mathematical experiments and mathematical science centers have a great impact on a mathematical education of the general public .

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Gabriele Kaiser

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Helen Forgasz

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Mellony Graven

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Alain Kuzniak

University of Alberta, Sherwood Park, Alberta, Canada

Elaine Simmt

East China Normal University, Shanghai, China

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Beutelspacher, A. (2018). Mathematical Experiments—An Ideal First Step into Mathematics. In: Kaiser, G., Forgasz, H., Graven, M., Kuzniak, A., Simmt, E., Xu, B. (eds) Invited Lectures from the 13th International Congress on Mathematical Education. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-72170-5_2

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Random Experiments

We may perform various activities in our daily existence, sometimes repeating the same actions though we get the same result every time. Suppose, in mathematics, we can directly say that the sum of all interior angles of a given quadrilateral is 360 degrees, even if we don’t know the type of quadrilateral and the measure of each internal angle. Also, we might perform several experimental activities, where the result may or may not be the same even when they are repeated under the same conditions. For example, when we toss a coin, it may turn up a tail or a head, but we are unsure which results will be obtained. These types of experiments are called random experiments.

Random Experiment in Probability

An activity that produces a result or an outcome is called an experiment. It is an element of uncertainty as to which one of these occurs when we perform an activity or experiment. Usually, we may get a different number of outcomes from an experiment. However, when an experiment satisfies the following two conditions, it is called a random experiment.

(i) It has more than one possible outcome.

(ii) It is not possible to predict the outcome in advance.

Let’s have a look at the terms involved in random experiments which we use frequently in probability theory. Also, these terms are used to describe whether an experiment is random or not.

Learn more about sample space here.

What is a Random Experiment?

Based on the definition of random experiment we can identify whether the given experiment is random or not. Go through the examples to understand what is a random experiment and what is not a random experiment.

Is picking a card from a well-shuffled deck of cards a random experiment?

We know that a deck contains 52 cards, and each of these cards has an equal chance to be selected.

(i) The experiment can be repeated since we can shuffle the deck of cards every time before picking a card and there are 52 possible outcomes.

(ii) It is possible to pick any of the 52 cards, and hence the outcome is not predictable before.

Thus, the given activity satisfies the two conditions of being a random experiment.

Hence, this is a random experiment.

Consider the experiment of dividing 36 by 4 using a calculator. Check whether it is a random experiment or not.

(i) This activity can be repeated under identical conditions though it has only one possible result.

(ii) The outcome is always 9, which means we can predict the outcome each time we repeat the operation.

Hence, the given activity is not a random experiment.

Examples of Random Experiments

Below are the examples of random experiments and the corresponding sample space.

Number of possible outcomes = 8

Number of possible outcomes = 36

Number of possible outcomes = 100

Similarly, we can write several examples which can be treated as random experiments.

Playing Cards

Probability theory is the systematic consideration of outcomes of a random experiment. As defined above, some of the experiments include rolling a die, tossing coins, and so on. There is another experiment of playing cards. Here, a deck of cards is considered as the sample space. For example, picking a black card from a well-shuffled deck is also considered an event of the experiment, where shuffling cards is treater as the experiment of probability.

A deck contains 52 cards, 26 are black, and 16 are red.

However, these playing cards are classified into 4 suits, namely Spades, Hearts, Diamonds, and Clubs. Each of these four suits contains 13 cards.

We can also classify the playing cards into 3 categories as:

Aces:  A deck contains 4 Aces, of which 1 of every suit. 

Face cards:  Kings, Queens, and Jacks in all four suits, also known as court cards.

Number cards:  All cards from 2 to 10 in any suit are called the number cards. 

• Spades and Clubs are black cards, whereas Hearts and Diamonds are red.
• 13 cards of each suit = 1 Ace + 3 face cards + 9 number cards
• The probability of drawing any card will always lie between 0 and 1.
• The number of spades, hearts, diamonds, and clubs is the same in every pack of 52 playing cards.

An example problem on picking a card from a deck is given above.

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Advancing the quest for dark matter: new insights from the lux-zeplin experiment.

Faculty and student researchers from Brown contributed key expertise on the LZ team’s latest findings, refining the search for dark matter particles and pushing the boundaries of detection technology.

Powerful light sensors assembled at Brown into two large arrays will keep watch on the LUX-ZEPLIN dark matter detector for the tell-tale flashes of dark matter particles. Photo by Matthew Kapust/Sanford Underground Research Facility

PROVIDENCE, R.I. [Brown University] — Dark matter is one of the biggest mysteries in physics and cosmology. Though it can't be seen directly, scientists suspect it’s what holds galaxies together and makes up about 85% of all mass in the universe.

A global team of over 250 scientists and engineers from 38 institutions — led by Lawrence Berkeley National Laboratory and including students, postdoctoral researchers and faculty from Brown University — are part of a worldwide effort to understand the true nature of dark matter and catch a glimpse of this elusive substance for the first time. They use a highly sensitive detector to look for tiny flashes of light that might occur when dark matter particles interact with atoms in the detector. While they don’t reveal the existence of dark matter, their latest results show that the team has been getting closer than any team of scientists to date.

Presented on Monday, Aug. 26, at conferences in Chicago and São Paulo, Brazil, the new results offer important insights into WIMPs, which are the leading theoretical candidates for comprising dark matter. Short for weakly interacting massive particles, WIMPs can’t be seen because they don’t absorb, emit or reflect light. And they interact with normal matter only on rare occasions, which is why they’re so hard to detect even when millions may be traveling through the Earth and everything on it each second.

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“Our goal is to improve our search even further with an even better level of sensitivity,” said LZ experiment cofounder  Richard Gaitskell , a Brown professor of physics and director of the  Center for the Fundamental Physics of the Universe . “The detector is very ready to see a signal.”

The LZ central detector in the clean room at Sanford Lab before beginning its journey underground. Photo by Matthew Kapust, Sanford Underground Research Facility.

The findings, while incremental, represent a significant advance, Gaitskell added. They help researchers narrow their search, improving their understanding of dark matter, if it exists. For instance, the experiment’s ability to detect extremely faint interactions now allows researchers to rule out many possible dark matter models that predict interactions stronger than what the data shows, ultimately dwindling down the options for where WIMPs could be hiding.

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Zeroing in on dark matter

The LZ detector is located a mile below the Black Hills of South Dakota in the Sanford Underground Research Facility. The descent into the former gold mine, which was also home to the precursor LUX experiment until 2016, takes about 10 minutes via elevator. The heart of the LZ consists of two nested titanium tanks filled with about 10 tons of pure liquid xenon. The tanks are monitored by photomultiplier tubes (PMTs) that will detect the dark matter particles if they’re there.

The theory for this detection works like this: If WIMPs are present, they may occasionally collide with the nucleus of a xenon atom, causing a tiny flash of light and some movement in the atoms. If that happens, the PMTs are meant to catch that interaction. Capturing that glimpse, of course, is exceedingly difficult and why most researchers instead infer things about dark matter through how its powerful gravity bends and focuses the light around it, a phenomenon called gravitational lensing.

Much work done on the two massive photodetector arrays happened in Barus and Holley, including building, testing and integrating more than 14,000 components. Photo by Nick Dentamaro.

In 2022, the detector passed a check-out phase of startup operations and delivered its first results , proving itself to be world’s most sensitive detector of dark matter and placing what were until now the strictest limits on how strongly WIMPs should interact with ordinary matter. To get the latest result, the analysis combined 220 days of new data taken in 2023 and 2024 with 60 days from the experiment's first run, which started in 2022. LZ is still in its early phases, too. By 2028, the LZ team plans to gather over 1,000 days of data.

“This is how science works,” Gaitskell said. “This is a huge question we are trying to answer: ‘What is most of the matter in the universe made of?’ It won’t get answered in a matter of weeks, months or even years necessarily. This is one that will take decades to answer, and that is actually pretty typical of most major scientific questions. If you go back and look at the history books, you'll realize that major discoveries are separated by long periods of time and happened step-by-step.”

This new results marks the first time that LZ has applied “salting”— a technique that adds fake WIMP signals during data collection. By camouflaging the real data until “unsalting” at the very end, researchers can avoid unconscious bias and keep from overly interpreting or changing their analysis before knowing the outcome.

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Brown’s role in the experiment

Since its start, early-career researchers have played a significant role in building and operating the experiment. Brown’s team, for instance, was crucial in building the PMT arrays , which serve as the experiment’s “eyes.”

The group worked with Berkeley Lab and Imperial College London researchers to design, test and assemble the arrays and its more than 14,000 components in cleanrooms at Brown’s Department of Physics, where the components also underwent two years of testing.

A photomultiplier tube is carefully inserted into the array inside the PMT Array Lifting And Commissioning Enclosure. Photo by Nick Dentamaro.

“The work didn’t stop once we started operating, either,” Gaitskell said. “Brown researchers have been directly responsible for continuing to run the systems that we built and oversee its care.”

This often involves hands-on work at the site itself. Brown physics Ph.D. student Chen Ding, for instance, spent three weeks in South Dakota performing crucial calibration tasks last summer. The experience was surreal, he said.

“Once you enter the lab, it's like a normal lab, but it's a very special experience getting there,” Ding said. “It's not just going down the elevator for 10 minutes — you still need to walk what feels like a mile down there and take a little train, all while there’s other excavation going on, and you’re in full safety gear.”

Another Brown Ph.D. student, Austin Vaitkus, who monitors the PMTs' performance, said the experience is invaluable to researchers, like him, at the start of their career. “When you are working on an experiment like this, you are working with the best people in the world on this subject,” Vaitkus said. “It’s really exciting to be able to say that you’re on a team that’s the best at doing what they do.”

Brown doctoral candidate Jihyeun Bang shares that enthusiasm, emphasizing the broader impact of the work: “Dark matter is one of the biggest challenges in the field, and it’s been motivating to see my work directly contribute to that overall goal.”

LZ is supported by the U.S. Department of Energy, the Science and Technology Facilities Council of the U.K., the Portuguese Foundation for Science and Technology, the Swiss National Science Foundation, and the Institute for Basic Science in Korea, with assistance from the Sanford Underground Research Facility.

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