what is experimental in math

Experimental Probability

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

What is Experimental Probability?

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

Experimental Probability Formula

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

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

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

Experimental Probability vs Theoretical Probability

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

Experimental Probability Examples

Here are a few examples from real-life scenarios.

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

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

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

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

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

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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.

Experimental Probability

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Experimental Probability vs. Theoretical Probability

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

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

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

Theoretical probability for Event A can be calculated as follows:

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

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

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

Experimental Probability: Examples

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

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

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

             $= \frac{4}{10}$

             $= \frac{2}{5}$

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

The experimental probability of rolling a 2 

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

$= \frac{5}{20}$

$= \frac{1}{4}$

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

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

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

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

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

               $= \frac{10}{25}$

               $= \frac{2}{5}$

               $= 0.4$

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

Solution: 

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

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

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

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

Number of times 3 showed $= 7$

Number of tosses $= 30$

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

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

Solution:  

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

So, the number of trials $= 20$

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

$= \frac{4}{5}$

$= 0.8$ or $80%$

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

Number of white bikes $= 100$ 

Total number of bikes $= 500$

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

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

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

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

What is the importance of experimental probability?

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

Is experimental probability always accurate?

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

Can experimental probability change every time the experiment is performed?

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

What is theoretical probability?

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

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Experimental Mathematics

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

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

1. Gaining insight and intuition.

2. Discovering new patterns and relationships.

3. Using graphical displays to suggest underlying mathematical principles.

4. Testing and especially falsifying conjectures.

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

6. Suggesting approaches for a formal proof.

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

8. Confirming analytically derived results.

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

Explore with Wolfram|Alpha

More things to try:

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• linear independence (a,b,c,d), (e,f,g,h), (i,j,k,l)

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

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Experimental probability

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

Experimental probability formula

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

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

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

The number of times an event occurred is p  = 20 

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

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

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

Difference between experimental probability and theoretical probability

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

Using a coin 

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

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

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

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

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

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

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

Using a spinner

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

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

1/4 = 0.25 = 25%

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

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

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

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

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

Theoretical probability

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Theoretical vs. Experimental Probability: How do they differ?

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

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

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

Table of Contents

What Is Theoretical Probability?

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

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

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

How Do You Calculate Theoretical Probability?

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

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

How Is Theoretical Probability Used in Real Life?

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

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

What Is Experimental Probability?

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

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

How Do You Calculate Experimental Probability?

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

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

How Is Experimental Probability Used in Real Life?

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

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

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

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

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

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

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

What to read next:

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

Wrapping Up

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

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

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Experimental mathematics

The advent of powerful computers enables mathematicians to look for patterns, correspondences, in fact, make up conjectures which have been verified in several computable cases. This activity is often referred to as "experimental mathematics" . By its very nature, this activity cannot be reported on in the rigorous theorem-proof style of exposition which is standard in mathematics. For this reason, experimental mathematics has created some interest groups of its own, e.g., around the Journal "Experimental Mathematics" , and in mathematical programming .

Yet, the possibility of performing more and more advanced computations by means of little human (programming) effort also had its effect on the standard mathematical rigour of exposition: nowadays conjectures are verified to rather a significantly high degree of computational complexity before being brought forward as such. See [a3] , [a1] for more applied examples.

Two examples in pure mathematics are the " Moonshine conjectures " , where modular functions are related to representations of sporadic finite simple groups (cf. [a2] ), and the "GUE hypothesis" , where joint distributions of zeros of the Riemann zeta-function are equated to those of the eigenvalues of matrices from GUE, the Gaussian unitary ensemble of large-dimensional random Hermitian matrices (cf. [a4] ).

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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.

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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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What is experimental probability? 

Practice questions, experimental probability – explanation & examples.

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

In this lesson we will go through:

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

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

Experimental Probability can be expressed mathematically as: 

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

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

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

How do we find experimental probability?

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

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

Let’s go through some examples. 

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

Number of coins showing Heads: 12

Total number of coins flipped: 20

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

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

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

b. What was the probability of a prime number?

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

a. Number of times 3 showed = 7

Number of tosses = 30

$P(\text{3}) = \frac{7}{30}$ 

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

Number of trials = 30 

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

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

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

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

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

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

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

b.  We expect the employee to be absent

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

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

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

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

a. Number of red cars = 50 

Total number of cars = 500 

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

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

Now it is time for you to try these examples. 

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

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

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

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

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

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

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

Number of brown jeans = 25

Total Number of jeans = 125

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

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

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

Number of beef burgers = 110 

Number of burgers (or sandwiches) sold = 200 

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

a. Number of comedian winners = 3

Number of seasons = 20 

$P(\text{comedian}) = \frac{3}{20}$ 

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

Number of winners that are dancers = 2 

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

Therefore we expect 

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

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

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

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Experimental Probability – Definition with Examples

Updated on January 9, 2024

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

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

What Is Experimental Probability?

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

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

Definition of Experimental Probability

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

Calculating Experimental Probability

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

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

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

Examples of Experimental Probability

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

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

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

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

Properties of Experimental Probability

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

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

Key Factors Affecting Experimental Probability

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

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

Difference Between Experimental and Theoretical Probability

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

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

Formulas for Calculating Experimental Probability

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

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

Writing Formulas for Experimental Probability

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

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

P(Heart) = 13/52 ≈ 0.25 = 25%

Use Cases of Experimental Probability in Real Life

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

Practice Problems on Experimental Probability

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

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

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

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

Frequently Asked Questions on Experimental Probability

What is the formula for experimental probability.

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

How is experimental probability calculated?

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

What is the difference between experimental and theoretical probability?

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

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

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

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Experimental Probability

The outcome of an actual experiment involving numerous trials is called experimental probability. Learn more about exper imental probability and its properties in this article. ...Read More Read Less

About Experimental Probability

Defining Probability

How precisely do we define experimental probability, formulation.

• Solved Examples
• Frequently Asked Questions

The mathematics of chance is known as probability (p). The probability of occurrence of an event (E) is revealed by probability.

The probability of an event can be expressed as a number between 0 and 1. 

The likelihood of an impossibility is zero. A probability between 0 and 1 can be attributed to any other events that fall in between these two extremes. Experimental probability is the probability that is established based on the outcomes of an experiment. The term ‘ empirical probability ’ is also used for the same concept.

A probability that has been established by a series of tests is called an experimental probability. To ascertain their possibility, a random experiment is conducted and iterated over a number of times; each iteration is referred to as a trial . 

The goal of the experiment is to determine the likelihood of an event occurring or not. 

It could involve spinning a spinner, tossing a coin, or using a dice. The probability of an event is defined mathematically as the number of occurrences of the event divided by the total number of trials.

The number of times an event occurred during the experiment divided by all the times the experiment was run is known as the experimental probability of that event. Each potential result is unknown, and the collection of all potential results is referred to as the sample space . 

Experimental probability is calculated using the following formula:

\(P(E)=\frac{Number~of~times~an~event~occurred~during~an~experiment}{The~total~number~of~times~the~experiment~was~conducte}\)

\(P(E)=\frac{n(E)}{n(S)}\)

n(E) = Number of events occurred

n(S) = Number of sample space

Solved Experimental Probability Examples

Example 1: The owner of a cake store is curious about the percentage of sales of his new gluten-free cupcake line. He counts the number of cakes that were sold on one day of the week, Monday, where he sold 30 regular and 70 gluten free cakes. Calculate the probability in this case.

According to the details in the question, the number of gluten free cakes is n(E) = 70 cakes.

Total number of cakes n(S) = 30 + 70 = 100 cakes.

Substituting these values in the formula.

\(P(E)=\frac{n(E)}{n(S)}\) = \(\frac{70}{100}\)   = 0.7 = 70%

Hence, the owner of the cake store finds that the gluten-free cupcakes will probably make up 70% of his weekly sales.

Example 2: A baseball manager is interested to know the probability that a prospective new player will hit a home run in the game’s first at-bat. The player has 11 home runs in 1921 games throughout his career. Calculate the probability of the player hitting a home run.

The data provided is, the player has hit 11 home runs, n(E) = 11

Total number of games, n(s) = 1921 games.

\(P(E)=\frac{n(E)}{n(S)}\) = \(\frac{11}{1921}\)   = 0.005726 = 0.5726%

He will therefore have a 0.5726 percent chance of hitting a home run in his first at-bat.

Example 3: A vegetable gardener is checking the likelihood that a fresh bitter gourd seed would germinate. He plants 100 seeds, and 57 of them sprout new plants. Calculate the probability in this scenario.

According to the question, the number of bitter gourd plants that sprouted is n(E) = 57.

Total number of seeds sown, n(S) = 100 seeds.

\(P(E)=\frac{n(E)}{n(S)}\) = \(\frac{57}{100}\)   = 0.57 = 57%

Hence, the probability that a new bitter gourd seed will be sprout is 57% . 

Example 4: Joe’s Bagel Shop sold 26 bagels in one day, 9 of which were raisin bagels. Calculate the percentage of raisin bagels that will be sold the following day using experimental probability.

As stated in the question, the number of raisin bagels, n(E) = 9.

Total number of bagels Joe sold, n(s) = 26 .

\(P(E)=\frac{n(E)}{n(S)}\) = \(\frac{9}{26}\)   = 0.346 = 34.6%

As a result, there is a 34.6 percent chance that Joe will sell raisin bagels the following day.

Do you simplify the probabilities of experiments?

Yes, the ratio obtained is simplified after the ratio between the frequency of the occurrence and the total number of trials is determined.

Which type of probability — theoretical or experimental — is more accurate?

Compared to experimental probability, theoretical probability is more precise. Only if there are more trials, then the results of experimental probability will be close to the results from theoretical probability.

How can experimental probability be calculated?

Actual tests and recordings of events serve as the foundation for calculating the experimental probability of an event. It is determined by dividing the total number of trials by the number of times an event occurred.

What is the chance of getting a 1 when you throw a dice?

A ‘1’ has a 1/6 experimental probability of rolling. Six faces, numbered from 1 to 6, make up a dice. Any number between 1 and 6 can be obtained by rolling the dice, and the likelihood of getting a 1 is equal to the ratio of favorable results to all other potential outcomes, or 1/6.

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What is an Experimental Math Course and Why Should We Care?

By: Lara Pudwell, Valparaiso University

What is the first meaningful mathematics problem you remember solving? For me, it was the nine dots, four lines puzzle . When my fourth grade teacher assigned it as an extra credit problem, I spent several days of recess scribbling out attempted solutions in the sandpit, erasing, and trying again until, at last, I found a solution!

I believe this geometric puzzle still sticks out in my memory nearly three decades later because it was one of the first experiences I had with trying to answer a question that didn’t simply involve mimicking previous work. For practitioners, informed trial-and-error is a key step in doing mathematics, so the idea of “thinking out of the box’’ (or in my case, literally thinking in the sandbox…) to build intuition seems natural. However, this is a far stretch from the view of many students who see mathematics as an opportunity to memorize formulas and execute repetitive tasks.

Where do students learn the process of refining mathematical conjectures? Certainly, teaching (via) inquiry in the mathematics classroom has generated much discussion, but often the conversation about inquiry is attached to particular material in the curriculum, with an inquiry-based approach to calculus or statistics, for example. Despite being fundamental to doing mathematics, the majority of the time the inquiry process is a means to an end, rather than a focus of an entire class, and it’s rarely addressed directly. In this environment, some students internalize the inquiry process by indirect exposure. Others finish their education without a true sense of how mathematics is actually developed.

Experimental mathematics courses are one answer to the need to celebrate and study inquiry for the sake of inquiry.  In particular, an experimental mathematics course is not a course about a particular set of material; it is a course about a particular approach to doing mathematics.

Courses in experimental mathematics have been offered by at least 7 different colleges and universities [1].  Outside of those who have taught or taken these courses, there is not widespread understanding of what “experimental mathematics” means in the undergraduate curriculum. My goal in this post is to give a better idea of what such a course looks like.

Comparing the syllabi of various experimental mathematics courses quickly shows the material isn’t standardized (nor does it need to be). However, these courses have some common themes.

First, students gain experience with programming and/or computer algebra systems throughout the course. While computer work is common to most experimental courses, it is not a necessary feature. Experimental mathematics could happen even without a computer in the room. (My solution to the nine dots, four lines problem occurred in a sandbox, rather than in a computer lab, and it still used experimental techniques to hone in on a solution.) However, in many cases, especially involving functions, number theory, combinatorics, and more, the use of a computer accomplishes the same thing one might do by hand in considerably less time. The computer can generate data and help sift through the results, quickly locating an example or counterexample. The goal, then, of using computers in an experimental math class is not just for the sake of using computers. Machine computation is a tool to greatly expand students’ reach as they explore.

Further, students build intuition via experimentation and conjecture, and, most importantly, students produce projects where they develop their own solutions to open-ended mathematics questions.

Experiment, conjecture, repeat

It’s tempting to say “my students experiment when I introduce a new concept with a group activity.” But the key word in that sentence is “introduce”. In the traditional syllabus, the focus is on material. Students can learn the material in a variety of ways, but a calculus syllabus is generally less focused on how students learn and more on the fact that they should learn about limits, derivatives, integrals, and their applications. Even if it is introduced in an interactive inquiry-based fashion, the star is the content.   When preparing for course assessments, students don’t study strategies for building intuition; they study the theorems and computations that class activities led them toward.

For example, in a calculus class, students learn the limit definition of the derivative. However, once they learn the conceptual idea that derivatives compute slope or rate of change, they look for speedups. We compute the derivatives of \(x^0\), \(x^1\), \(x^2\), and \(x^3\) using the limit definition and look for a pattern.  State the pattern and practice computing with other functions like \(x^7\), \(x^{42}\), \(x^{1/2}\), or \(x^{-5}\).  This approach is computational.  In fact, it can be done in a way that builds intuition and uses active learning pedagogy.  But this is still a content-centered lesson.  The takeaway: \(d/dx(x^n)=nx^{n-1}\).

In experimental mathematics, the star is the approach to new information. The content could be different each time the course is taught, but the method of figuring out new information involves a sequence of experiment, conjecture, and repeat.

A “typical” experimental math class meeting might look something like this:

1.  The instructor presents a new mathematics problem, leading class discussion long enough to make the problem statement clear.

For example, one topic that lends itself well to exploration with experimental methodology is continued fractions. An entire class can be built on the idea that any real number \(r\) can be written as \(r=a_0+1/(a_1+1/(a_2+1/\ldots))\), where the number of integers \(a_i\) required to write \(r\) could be finite or infinite. The instructor presents the definition of continued fraction and computes the continued fraction form of a few well-chosen real numbers by hand. Then students are asked to look for (families of) real numbers whose continued fraction expansions have predictable structure.

2.  Students brainstorm in small groups to determine what data might help them better solve the problem and then gather the data, often with computer assistance.

In this case, students will find it useful to write code that inputs a real number \(r\) and outputs the first \(n\) terms \(a_0, a_1, a_2, \dots\) in the continued fraction expansion for \(r\). They may also find it helpful to input the terms \(a_0, a_1, a_2, \dots\) in a continued fraction expansion and output the simplified corresponding real number. Students use the data to conjecture a solution or patterns for special cases.  At this point, the experimentation begins in earnest.

Clearly any finite continued fraction represents a rational number, but is the converse true? What patterns are there in the continued fractions for irrational numbers? Students play with expansions for\(\pi\), \(e\), and powers of \(\pi\) and \(e\); \(e^n\) has some nice structure when \(n\) is an integer that \(\pi^n\) does not. The continued fraction for \(\sqrt{n}\) (where \(n\) is a positive integer) has particularly attractive eventually-periodic structure. The instructor’s role during this exploration can vary. I circulate around the room and talk with individual students as they work. I also periodically ask students to share interesting observations with the rest of the class. Then, if one person hits on a promising idea, it can quickly be shared across the room, encouraging other students to explore related avenues of inquiry.

3.  If possible, students prove their conjectures. If not, students refine or revise their conjectures by iterating steps (2)-(3). Or students try the same process for a related or generalized version of the same problem.

In this case, students may notice the eventually-periodic structure of the continued fraction for \(\sqrt{n}\) and conjecture patterns for entire families of continued fractions for square roots. This could lead students to ask a converse question: if I have a periodic continued fraction, is it necessarily the continued fraction for a square root? If so, which one? If not, what other kinds of numbers have periodic continued fractions? Students could also look at generalized continued fraction expansions, where the numerators in the fractions aren’t all 1s.

The big idea from this class meeting is not “square roots have periodic continued fractions”. The takeaway is: you made a conjecture and confirmed or refined it; what’s the next natural conjecture? The instructions given at the beginning of class have some specific mathematical questions to get started, but they also involve reflection. For example, the final part of students’ written work from class could be to write a paragraph response to: “if you were to continue this problem, what question would you investigate next and why?” The particular material is less important than the process of asking and revising questions.

Classrooms require structure. Often, that course structure is dictated by a list of content and computational skills that are required in subsequent courses. Many days of experimental mathematics courses also require structure so that student learning can be assessed. In both the calculus example and the continued fraction example above, one could argue that structure exists because the instructor has a clear end result in mind, whether students take different routes or a prescribed route to get there. However, it is also possible to provide structure without having a pre-determined final content goal in mind.

The beauty of experimental mathematics is that it gives students the tools to conduct open-ended inquiry, which is often described in terms of a “project”. These could consist of several weekly projects or could take the form of a single long-term project. Projects involve questions where a student can’t just conduct a literature search and find answers to all the questions they generate. On the other hand, it’s ok if their work isn’t all new to the literature, but has some overlap of re-discovered results. In my experimental mathematics class, I have students complete one large, semester-long project. I make sure that no two students have the same project area on any given iteration of the course. There is a project deadline approximately once per month during the four-month semester.

• Month one: Pick a project topic. Each student selects a broad area they want to investigate, but not necessarily a specific research question yet.
• Month two: Each student presents a question or two they’ve independently decided to explore and gets feedback from their classmates.
• Month three: Each student submits a preliminary written report on their progress to me for feedback.
• Month four: Each student gives a 15-minute conference-style presentation on the results of their project to the rest of the class.

At the beginning of each semester, I provide a list of suggested project areas, including some resources for finding tractable open problems. Students may choose from the list, propose a twist on a suggested project, or propose their own problem. The delightful thing about these projects is: in the four iterations of the course I’ve taught at Valparaiso (representing over 40 different student projects), some general topics have been selected more than once, but each time, the students went in completely different directions with their experimentation.   For example, several students have chosen to study cellular automata, but one student may look at variations of rules that generate the automata and study entire families of automata, while another student may become very interested in a particular automaton and study iterations of that one automaton over time. Either way, by the end of the course, each student has true ownership of their project and the direction it took over the duration of the semester. While they’ve had me and their classmates as a sounding board, the final result of the project was never prescribed to them. It is the result of doing what we do each day in class: program, experiment, conjecture, and repeat, run over the course of an entire semester to see what happens.

As long as we only discuss inquiry in the context of standard course material, we’re missing half of mathematics! The true joy of doing research is trying a problem that has never been solved before and then experimenting and refining conjectures until you hone in on something that works. Experimental mathematics lets students experience the process of how mathematics is actually discovered. Far beyond generating examples together or following inquiry-based activities to arrive at an expected theorem, experimental mathematics pushes discussion and inquiry past the standard classroom boundaries and owns up to the process of making your own conjectures without an intended final answer. Ultimately, students will be richer when we encourage them to create and answer their own questions instead of only leading them through our own.

[1] Links to experimental math courses:

Dartmouth College ( https://math.dartmouth.edu/archive/m56s13/public_html/ ) Grinnell College ( http://www.math.grin.edu/~chamberl/courses/444/syllabus.html ) Ithaca College ( http://www.tandfonline.com/doi/abs/10.1080/10511970.2013.870264 ) Lynchburg College ( http://lasi.lynchburg.edu/peterson_km/public/Courses/Fall%202016/Math350_f16.htm ) Rutgers University ( http://www.math.rutgers.edu/~zeilberg/math611.html ) Tulane University ( http://129.81.170.14/~vhm/syllabus.pdf ) Valparaiso University ( http://www.tandfonline.com/doi/abs/10.1080/10511970.2016.1143899?journalCode=upri20

2 Responses to What is an Experimental Math Course and Why Should We Care?

Great post! Agree! The Arnol’d book “Experimental Math” is a great book promoting similar ideas.

Yes! www.artofmathematics.org has a lot of explorations in the same spirit 🙂 We use them for math and non-math majors.

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Title: s$^3$c-math: spontaneous step-level self-correction makes large language models better mathematical reasoners.

Abstract: Self-correction is a novel method that can stimulate the potential reasoning abilities of large language models (LLMs). It involves detecting and correcting errors during the inference process when LLMs solve reasoning problems. However, recent works do not regard self-correction as a spontaneous and intrinsic capability of LLMs. Instead, such correction is achieved through post-hoc generation, external knowledge introduction, multi-model collaboration, and similar techniques. In this paper, we propose a series of mathematical LLMs called S$^3$c-Math, which are able to perform Spontaneous Step-level Self-correction for Mathematical reasoning. This capability helps LLMs to recognize whether their ongoing inference tends to contain errors and simultaneously correct these errors to produce a more reliable response. We proposed a method, which employs a step-level sampling approach to construct step-wise self-correction data for achieving such ability. Additionally, we implement a training strategy that uses above constructed data to equip LLMs with spontaneous step-level self-correction capacities. Our data and methods have been demonstrated to be effective across various foundation LLMs, consistently showing significant progress in evaluations on GSM8K, MATH, and other mathematical benchmarks. To the best of our knowledge, we are the first to introduce the spontaneous step-level self-correction ability of LLMs in mathematical reasoning.

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