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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.

I am Altiné. I am the guy behind mathodics.com. When I am not teaching math, you can find me reading, running, biking, or doing anything that allows me to enjoy nature's beauty. I hope you find what you are looking for while visiting mathodics.com.

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Percent error formula.

When you calculate results that are aiming for known values, the percent error formula is useful tool for determining the precision of your calculations. The formula is given by:

The experimental value is your calculated value, and the actual value is the known value (sometimes called the accepted or theoretical value). A percentage very close to zero means you are very close to your targeted value, which is good. It is always necessary to understand the cause of the error, such as whether it is due to the imprecision of your equipment, your own estimations, or a mistake in your experiment.

Example Question

The 17th century Danish astronomer, Ole Rømer, observed that the length of the eclipses of Jupiter by its satellites would appear to fluctuate depending on the direction Earth was traveling relative to Jupiter at the time of the eclipse. If Earth was traveling toward Jupiter, the eclipes of Jupiter by, say, Io, would last for a shorter amount of time, while if Earth was traveling away from Jupiter, the eclipses would appear to be longer. In 1676, he determined that this phenomenon was due to the fact that the speed of light was finite, and subsequently estimated its velocity to be approximately 220,000 km/s. The current accepted value of the speed of light is almost 299,800 km/s. What was the percent error of Rømer's estimate?

Experimental value = 220,000 km/s = 2.2 x 10 8 m/s

Actual value = 299,800 km/s = 2.998 x 10 8 m/s

So Rømer was quite a bit off by our standards today, but considering he came up with this estimate at a time when a majority of respected astronomers, like Cassini, still believed that the speed of light was infinite, his conclusion was an outstanding contribution to the field of Astronomy.

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Comparing experimental and theoretical probability

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• Experimental probability VS. Theoretical probability

• What is the experimental probability of both coins landing on heads?
• Calculate the theoretical probability of both coins landing on heads.
• Compare the theoretical probability and experimental probability.
• What can Jessie do to decrease the difference between the theoretical probability and experimental probability?

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Topic Notes

Introduction.

Comparing experimental and theoretical probability is a fundamental concept in statistics that helps us understand the relationship between predicted outcomes and real-world results. Our introduction video provides a comprehensive overview of this topic, serving as an essential starting point for students and enthusiasts alike. This article delves deeper into the subject, exploring the definitions, key differences, and practical applications of both theoretical and experimental probability. Theoretical probability is based on mathematical calculations and assumes ideal conditions, while experimental probability relies on actual observations and data collection . Understanding the interplay between these two types of probability is crucial for making informed decisions in various fields, from scientific research to everyday life. As we progress through this article, we'll examine how these concepts are applied in real-world scenarios, highlighting their importance in fields such as finance, weather forecasting, and quality control. By the end, you'll have a solid grasp of how theoretical and experimental probability complement each other in our quest to understand and predict outcomes.

Defining Theoretical and Experimental Probability

Probability is a fundamental concept in mathematics that helps us understand the likelihood of events occurring. Two key types of probability are theoretical probability and experimental probability. Understanding these concepts is crucial for anyone studying statistics or working with data analysis.

Theoretical probability, also known as classical probability, is based on mathematical calculations and logical reasoning. It represents the expected likelihood of an event occurring under ideal conditions. This type of probability is determined by analyzing the possible outcomes of a situation and calculating the ratio of favorable outcomes to the total number of possible outcomes. Theoretical probability is often expressed as a fraction, decimal, or percentage.

On the other hand, experimental probability, sometimes called empirical probability, is derived from actual trials or experiments. It represents the observed frequency of an event occurring in real-world situations. Experimental probability is calculated by conducting multiple trials and recording the number of times a specific outcome occurs, then dividing that number by the total number of trials. Like theoretical probability, it can be expressed as a fraction, decimal, or percentage.

To illustrate these concepts, let's consider the classic example of a coin flip. In theory, a fair coin has two equally likely outcomes: heads or tails. The theoretical probability of getting heads on a single flip is 1/2 or 50%. This calculation is based on the assumption that the coin is perfectly balanced and the flip is unbiased.

However, when we actually flip a coin multiple times, we might observe slightly different results. For instance, if we flip a coin 100 times and get 48 heads, the experimental probability of getting heads would be 48/100 or 48%. This result is based on the actual outcomes of the experiment rather than theoretical calculations.

Calculating theoretical probability involves identifying all possible outcomes and determining the number of favorable outcomes. For example, when rolling a six-sided die, the theoretical probability of rolling an even number is 3/6 or 1/2, as there are three favorable outcomes (2, 4, and 6) out of six possible outcomes.

Experimental probability calculations require conducting multiple trials and recording the results. For instance, if we roll a die 200 times and observe 98 even numbers, the experimental probability would be 98/200 or 49%. This approach provides a real-world estimate of the probability based on observed data.

It's important to note that as the number of trials in an experiment increases, the experimental probability tends to converge towards the theoretical probability. This phenomenon is known as the law of large numbers. However, due to various factors such as imperfections in objects or experimental conditions, there may always be slight differences between theoretical and experimental probabilities.

Understanding both theoretical and experimental probability is essential in various fields, including statistics, science, and data analysis. Theoretical probability provides a baseline expectation, while experimental probability offers insights into real-world outcomes. By comparing these two types of probability, researchers can identify discrepancies, validate theories, and make informed decisions based on both mathematical models and empirical evidence.

Comparing Theoretical and Experimental Probability

Understanding the key differences between theoretical and experimental probability is crucial in the field of statistics and probability theory. Theoretical probability, also known as classical probability, is based on mathematical calculations and assumes ideal conditions. It represents the expected likelihood of an event occurring in the long run. On the other hand, experimental probability, or empirical probability, is derived from actual observations and experiments conducted in real-world scenarios.

The main distinction between these two types of probability lies in their approach and application. Theoretical probability is calculated using mathematical formulas and logical reasoning, without the need for physical experiments. It assumes that all outcomes are equally likely and that the conditions remain constant. For instance, when considering a fair six-sided die, the theoretical probability of rolling any number is 1/6, as there are six equally possible outcomes.

Experimental probability, however, is determined by conducting actual trials or experiments and recording the results. It takes into account real-world factors and variations that may influence the outcomes. This type of probability is based on the frequency of occurrences in a given number of trials. For example, if you roll a die 100 times and observe that the number 4 appears 18 times, the experimental probability of rolling a 4 would be 18/100 or 0.18.

One of the most intriguing aspects of probability is that experimental results might not always match theoretical expectations, especially when dealing with small sample sizes. This discrepancy can be attributed to several factors. Firstly, random chance plays a significant role in small samples, leading to potential deviations from the expected outcomes. Secondly, real-world conditions may introduce subtle biases or imperfections that are not accounted for in theoretical calculations.

To illustrate this point, let's consider the die rolling example from the video. In theory, each number on a fair die has an equal probability of 1/6 or approximately 0.167. However, if you were to roll the die only 10 times, you might observe results that differ significantly from this theoretical expectation. For instance, you might roll a 3 four times out of 10 rolls, giving an experimental probability of 0.4 for rolling a 3. This result clearly deviates from the theoretical probability of 0.167.

The reason for this discrepancy lies in the small sample size. With only 10 rolls, random fluctuations have a more pronounced effect on the results. It's important to note that such deviations do not necessarily indicate that the die is unfair or that the theoretical probability is incorrect. Instead, it highlights the impact of sample size on experimental probability.

As the number of trials increases, a fascinating phenomenon occurs: experimental probability tends to approach theoretical probability. This concept is known as the Law of Large Numbers. In our die rolling example, if you were to increase the number of rolls to 1,000 or even 10,000, you would likely observe that the experimental probabilities for each number start to converge towards the theoretical probability of 1/6.

This convergence happens because larger sample sizes provide more opportunities for random fluctuations to balance out. With more trials, the impact of short-term variations diminishes, and the overall pattern begins to align more closely with the expected theoretical outcomes. It's important to note that while experimental probability may approach theoretical probability with a large number of trials, it may never exactly match it due to real-world factors and inherent randomness.

Understanding the relationship between theoretical and experimental probability is crucial in various fields, including scientific research, quality control, and data analysis. It helps researchers and analysts interpret results more accurately, especially when dealing with small sample sizes. By recognizing the potential for discrepancies between theoretical expectations and experimental outcomes, one can avoid drawing hasty conclusions and instead consider the role of sample size and other influencing factors.

In conclusion, while theoretical and experimental probability offer different approaches to understanding the likelihood of events, they are complementary in nature. Theoretical probability provides a foundation for expectations, while experimental probability offers insights into real-world outcomes. By considering both aspects and understanding their relationship, we can gain a more comprehensive and nuanced understanding of probability in various contexts.

Calculating and Interpreting Probabilities

Understanding how to calculate both theoretical and experimental probabilities is crucial in various fields, from statistics to everyday decision-making. This guide will walk you through the process step-by-step, using practical examples to illustrate the concepts.

How to Calculate Theoretical Probability

Theoretical probability is based on the assumption that all outcomes are equally likely. To calculate it:

• Identify the total number of possible outcomes.
• Determine the number of favorable outcomes (those that meet your criteria).
• Divide the number of favorable outcomes by the total number of possible outcomes.

Example: Finding the probability of rolling an even number on a six-sided die.

• Total outcomes: 6 (1, 2, 3, 4, 5, 6)
• Favorable outcomes: 3 (2, 4, 6)
• Probability = 3/6 = 1/2 or 0.5 or 50%

How to Find Experimental Probability

Experimental probability is based on actual trials or observations. To calculate it:

• Conduct a series of trials or observations.
• Count the number of times the desired outcome occurs.
• Divide the number of favorable outcomes by the total number of trials.

Example: Rolling a die 100 times and counting even numbers.

• Total rolls: 100
• Number of even rolls: 48
• Experimental probability = 48/100 = 0.48 or 48%

Using Frequency Charts for Experimental Probability

Frequency charts for probability are valuable tools for determining experimental probability:

• Create a chart listing all possible outcomes.
• Tally the occurrences of each outcome during your trials.
• Calculate the probability by dividing each outcome's frequency by the total number of trials.

Example: A frequency chart for probability for 50 coin flips might show:

• Heads: 27 (Probability: 27/50 = 0.54)
• Tails: 23 (Probability: 23/50 = 0.46)

Interpreting Probability Results

Understanding what probability results mean is crucial:

• A probability of 0 means the event will never occur.
• A probability of 1 (or 100%) means the event will always occur.
• A probability of 0.5 (or 50%) means the event is equally likely to occur or not occur.

In practical terms, a 60% chance of rain means that, under similar conditions, it rains 60 out of 100 times. It doesn't guarantee rain but suggests it's more likely than not.

Comparing Theoretical and Experimental Probabilities

Often, theoretical and experimental probabilities differ slightly:

• Theoretical probability assumes ideal conditions.
• Experimental probability reflects real-world variations.
• As the number of trials increases, experimental probability typically approaches theoretical probability.

Practical Applications

Understanding probability is essential in various fields:

• Weather forecasting: Predicting the likelihood of specific weather conditions.

Advantages and Limitations of Theoretical and Experimental Probability

Probability is a fundamental concept in mathematics and statistics, with two primary approaches: theoretical and experimental probability. Each method has its own set of advantages and limitations, making them suitable for different situations. Understanding these differences is crucial for accurately predicting outcomes and making informed decisions.

Theoretical probability, also known as classical probability, is based on mathematical calculations and assumes ideal conditions. Its main advantage lies in its ability to provide precise predictions for well-defined scenarios. For instance, when calculating the probability of rolling a specific number on a fair die, theoretical probability offers an exact mathematical solution. This approach is particularly useful in games of chance, risk assessment, and scientific modeling where conditions can be controlled or standardized.

However, theoretical probability has limitations. It often assumes perfect conditions that may not exist in the real world. For example, a coin toss is theoretically a 50/50 chance, but factors like air resistance, the tosser's technique, or an uneven surface can influence the outcome. In complex real-world scenarios, theoretical probability may oversimplify the situation, leading to inaccurate predictions.

On the other hand, experimental probability, also called empirical probability, is based on actual observations and data collection . One of the key advantages of experimental probability is its ability to account for real-world factors that theoretical models might miss. This approach is particularly valuable in situations where conditions are not ideal or when multiple variables are at play. For instance, in weather forecasting, experimental probability based on historical data and current observations often provides more accurate predictions than purely theoretical models.

A significant advantage that experimental probability has over theoretical probability is its capacity to reveal unexpected patterns or influences that might not be apparent in a theoretical model. By conducting repeated trials and collecting data, researchers can uncover factors affecting outcomes that weren't initially considered. This makes experimental probability invaluable in fields like medicine, social sciences, and economics, where human behavior and complex interactions play a crucial role.

However, experimental probability is not without its limitations. It requires a large number of trials to produce reliable results, which can be time-consuming and costly. The accuracy of experimental probability also depends on the quality of data collection and the representativeness of the sample. Biases in sampling or measurement errors can lead to skewed results. Additionally, rare events may be underrepresented in experimental data, potentially leading to inaccurate probability estimates for uncommon occurrences.

In practice, the choice between theoretical and experimental probability often depends on the specific situation and available resources. Theoretical probability is preferred in scenarios where conditions can be well-defined and controlled, such as in casino games or basic physics problems. It's also useful for initial predictions or when conducting experiments is impractical or impossible.

Experimental probability, however, shines in complex, real-world applications where multiple factors influence outcomes. It's particularly valuable in fields like epidemiology, market research, and environmental science. In these areas, the ability to account for unforeseen variables and real-world conditions makes experimental probability a more reliable approach.

Ultimately, the most effective approach often involves combining both methods. Theoretical probability can provide a baseline expectation, while experimental probability can refine and validate these predictions in real-world contexts. This complementary use of both approaches allows for more robust and accurate probability assessments, leading to better decision-making and more reliable predictions across various fields of study and practical applications.

Practical Applications and Examples

Theoretical and experimental probability play crucial roles in various fields, from statistics and science to everyday decision-making. Let's explore some real-world examples and applications of these concepts, along with exercises to help you practice calculating probabilities.

Weather Forecasting

Meteorologists use both theoretical and experimental probability to predict weather patterns. Theoretical probability is applied when analyzing historical data and atmospheric models, while experimental probability comes into play as they gather real-time data from weather stations and satellites. For example, a meteorologist might state that there's a 70% chance of rain tomorrow based on a combination of these probabilities.

Sports Statistics

In sports analytics, theoretical probability is used to calculate the likelihood of certain outcomes based on past performance and team statistics. Experimental probability is then applied as games are played and new data is collected. For instance, a basketball player's free throw percentage is an example of experimental probability, while the theoretical probability of making a three-pointer might be calculated based on court position and defensive pressure.

Medical Research

In clinical trials, researchers use theoretical probability to design studies and determine sample sizes. As the trial progresses, they collect data and calculate experimental probabilities to assess the effectiveness of treatments. For example, the theoretical probability of a new drug's success might be estimated based on its chemical properties, while the experimental probability is determined through actual patient outcomes.

Insurance Industry

Insurance companies rely heavily on probability to set premiums and assess risk. They use theoretical probability models based on demographic data and historical trends, then adjust these models with experimental probability as they gather data from their policyholders. For instance, the probability of a car accident for a specific age group might be theoretically calculated, then refined with actual claim data.

Quality Control in Manufacturing

Manufacturers use probability to maintain product quality. Theoretical probability helps in setting quality standards, while experimental probability is used in sampling and testing products. For example, a theoretical model might predict the probability of defects in a production line, which is then compared to the actual defect rate found through quality control checks.

Financial Markets

Investors and financial analysts use probability to assess market trends and make investment decisions. Theoretical models predict stock performance based on economic indicators, while experimental probability is derived from actual market data. For instance, the probability of a stock price increase might be theoretically calculated, then compared to its actual performance over time.

1. Theoretical Probability: A fair six-sided die is rolled. Calculate the probability of rolling an even number.

2. Experimental Probability: In a bag of 100 marbles, you draw marbles 50 times with replacement and find that you've drawn a red marble 15 times. What is the experimental probability of drawing a red marble?

3. Combined Exercise: A weather forecast predicts a 30% chance of rain based on historical data. Over the next 10 days, it actually rains on 4 days. Compare the theoretical and experimental probabilities.

4. Real-world Application: A basketball player has made 80 out of 100 free throws this season. What is the experimental probability of them making their next free throw? How might this differ from the theoretical probability based on their career average?

By understanding and applying both theoretical and experimental probability, we can make more informed decisions in various aspects of life. Whether it's predicting outcomes, assessing risks, or analyzing data, probability serves as a powerful tool across numerous fields. Practice with these concepts will enhance your ability to interpret and use probability in real-world situations.

In this article, we've explored the fundamental concepts of theoretical and experimental probability, building upon the foundation laid in the introductory video. We've discussed the key differences between these two approaches and their practical applications in various fields. Understanding both theoretical calculations and real-world experimental results is crucial for a comprehensive grasp of probability. The examples provided illustrate how these concepts apply to everyday situations, from coin tosses to complex scientific experiments. As you continue your studies, remember to apply these principles to enhance your critical thinking and decision-making skills. Probability theory is not just an academic exercise but a powerful tool for navigating uncertainties in life. We encourage you to delve deeper into this fascinating subject, experiment with probability scenarios, and share your insights with others. For further exploration, consider joining online forums, attending workshops, or exploring advanced probability courses to expand your knowledge and practical skills in this essential mathematical field.

Step 1: Understanding the Problem

First, we need to understand the problem. Jessie flips two coins 20 times to determine the experimental probability of landing on heads versus tails. The results are given in a table format, showing the outcomes and the number of times each outcome occurred. Our task is to find the experimental probability of both coins landing on heads (H, H).

Step 2: Analyzing the Results

• H, H: 4 times
• H, T: 5 times
• T, H: 6 times
• T, T: 5 times

Step 3: Calculating the Experimental Probability

To calculate the experimental probability, we use the formula: Experimental Probability = (Number of Successful Outcomes) / (Total Number of Trials) In this case, the number of successful outcomes (both coins landing on heads) is 4, and the total number of trials is 20. Therefore, the experimental probability is: Experimental Probability = 4 / 20

Step 4: Simplifying the Fraction

To simplify the fraction, we find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 4 and 20 is 4. We divide both the numerator and the denominator by their GCD: 4 ÷ 4 = 1 20 ÷ 4 = 5 Therefore, the simplified fraction is: 1 / 5

Step 5: Interpreting the Result

The experimental probability of both coins landing on heads is 1 out of 5, or 1/5. This means that, based on Jessie's experiment, there is a 20% chance (since 1/5 = 0.20) that both coins will land on heads in any given trial.

Here are some frequently asked questions about theoretical and experimental probability:

1. What is the difference between theoretical and experimental probability?

Theoretical probability is based on mathematical calculations and assumes ideal conditions, while experimental probability is derived from actual observations and data collection. Theoretical probability predicts outcomes based on logical reasoning, whereas experimental probability reflects real-world results.

2. How do you calculate theoretical probability?

To calculate theoretical probability, use the formula: P(event) = (number of favorable outcomes) / (total number of possible outcomes). For example, the theoretical probability of rolling a 3 on a fair six-sided die is 1/6, as there is one favorable outcome out of six possible outcomes.

3. What is an example of experimental probability?

An example of experimental probability is flipping a coin 100 times and observing 52 heads. The experimental probability of getting heads would be 52/100 = 0.52 or 52%. This result is based on actual observations rather than theoretical calculations.

4. Why might theoretical and experimental probabilities differ?

Theoretical and experimental probabilities may differ due to factors such as sample size, random variation, and real-world conditions that aren't accounted for in theoretical models. As the number of trials increases, experimental probability tends to converge towards theoretical probability.

5. Which is more reliable, theoretical or experimental probability?

Neither is inherently more reliable; they serve different purposes. Theoretical probability provides a baseline expectation, while experimental probability reflects real-world outcomes. In practice, combining both approaches often yields the most comprehensive understanding of probability in a given situation.

Prerequisite Topics

Before delving into the intricacies of comparing experimental and theoretical probability, it's crucial to understand the foundational concepts that support this topic. One of the most important prerequisite topics is influencing factors in data collection . This fundamental concept plays a significant role in shaping our understanding of probability and its practical applications.

Understanding the influencing factors in data collection is essential when comparing experimental and theoretical probability. These factors directly impact the quality and reliability of the data we use to calculate experimental probability. For instance, when conducting experiments to determine probability, the methods used for data collection can significantly influence the outcomes. Biases, sample size, and data collection techniques all play crucial roles in shaping the experimental results.

Moreover, the concept of data collection is intrinsically linked to the process of gathering empirical evidence, which is the cornerstone of experimental probability. By understanding how various factors can influence data collection, students can better appreciate the potential discrepancies between experimental and theoretical probability.

For example, when comparing the theoretical probability of rolling a six on a die (1/6) with the experimental probability derived from actual rolls, the influencing factors in data collection become evident. The number of rolls (sample size), the randomness of the rolling technique, and even the physical characteristics of the die can all affect the experimental results. These factors highlight the importance of understanding data collection principles when interpreting and comparing probabilities.

Furthermore, grasping the concepts related to influencing factors in data collection helps students develop a critical eye when analyzing probability studies. It enables them to identify potential sources of error or bias in experimental setups, leading to a more nuanced understanding of the relationship between theoretical and experimental probabilities.

In conclusion, the prerequisite topic of influencing factors in data collection serves as a crucial foundation for comparing experimental and theoretical probability. It equips students with the necessary tools to critically evaluate data, understand the limitations of experimental results, and appreciate the complexities involved in real-world probability calculations. By mastering this prerequisite, students will be better prepared to tackle the challenges of comparing different types of probabilities and apply their knowledge in various statistical contexts.

Basic Concepts

• Introduction to probability
• Organizing outcomes
• Probability of independent events

Related Concepts

• Determining probabilities using tree diagrams and tables
• Probability

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AP®︎/College Statistics

Course: ap®︎/college statistics   >   unit 7.

• Intro to theoretical probability

Experimental versus theoretical probability simulation

• Theoretical and experimental probability: Coin flips and die rolls
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Calculate Percent Error 5

Percent Error Definition

Percent error, sometimes referred to as percentage error, is an expression of the difference between a measured value and the known or accepted value . It is often used in science to report the difference between experimental values and expected values.

The formula for calculating percent error is:

Note: occasionally, it is useful to know if the error is positive or negative. If you need to know the positive or negative error, this is done by dropping the absolute value brackets in the formula. In most cases, absolute error is fine. For example, in experiments involving yields in chemical reactions, it is unlikely you will obtain more product than theoretically possible.

Steps to Calculate the Percent Error

• Subtract the accepted value from the experimental value.
• Take the absolute value of step 1
• Divide that answer by the accepted value.
• Multiply that answer by 100 and add the % symbol to express the answer as a percentage .

Example Calculation

Now let’s try an example problem.

You are given a cube of pure copper. You measure the sides of the cube to find the volume and weigh it to find its mass. When you calculate the density using your measurements, you get 8.78 grams/cm 3 . Copper’s accepted density is 8.96 g/cm 3 . What is your percent error?

Solution: experimental value = 8.78 g/cm 3 accepted value = 8.96 g/cm 3

Step 1: Subtract the accepted value from the experimental value.

8.78 g/cm 3 – 8.96 g/cm 3 = -0.18 g/cm 3

Step 2: Take the absolute value of step 1

|-0.18 g/cm 3 | = 0.18 g/cm 3

Step 3: Divide that answer by the accepted value.

Step 4: Multiply that answer by 100 and add the % symbol to express the answer as a percentage.

0.02 x 100 = 2 2%

The percent error of your density calculation is 2%.

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5 thoughts on “ calculate percent error ”.

Percent error is always represented as a positive value. The difference between the actual and experimental value is always the absolute value of the difference. |Experimental-Actual|/Actualx100 so it doesn’t matter how you subtract. The result of the difference is positive and therefore the percent error is positive.

Percent error is always positive, but step one still contains the error initially flagged by Mark. The answer in that step should be negative:

experimental-accepted=error 8.78 – 8.96 = -0.18

In the article, the answer was edited to be correct (negative), but the values on the left are still not in the right order and don’t yield a negative answer as presented.

Mark is not correct. Percent error is always positive regardless of the values of the experimental and actual values. Please see my post to him.

Say if you wanted to find acceleration caused by gravity, the accepted value would be the acceleration caused by gravity on earth (9.81…), and the experimental value would be what you calculated gravity as :)

If you don’t have an accepted value, the way you express error depends on how you are making the measurement. If it’s a calculated value, like, based on a known about of carbon dioxide dissolved in water, then you have a theoretical value to use instead of the accepted value. If you are performing a chemical reaction to quantify the amount of carbonic acid, the accepted value is the theoretical value if the reaction goes to completion. If you are measuring the value using an instrument, you have uncertainty of the instrument (e.g., a pH meter that measures to the nearest 0.1 units). But, if you are taking measurements, most of the time, measure the concentration more than once, take the average value of your measurements, and use the average (mean) as your accepted value. Error gets complicated, since it also depends on instrument calibration and other factors.

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How to Calculate Percent Error

What Is the Formula for Percent Error?

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Percent error or percentage error expresses the difference between an approximate or measured value and an exact or known value as a percentage. It is a well-known type of error calculation, along with absolute and relative error.

Percent error plays a crucial role in validating hypotheses and assessing the accuracy of measurements in scientific research, and it also plays a fundamental role in quality control processes, where deviations from expected values could signify potential flaws in manufacturing or experimental procedures.

Here is the formula used to calculate percent error, along with an example calculation.

Key Points: Percent Error

• The purpose of a percent error calculation is to gauge how close a measured value is to a true value.
• Percent error is equal to the difference between an experimental and theoretical value, divided by the theoretical value, and then multiplied by 100 to give a percent.
• In some fields, percent error is always expressed as a positive number. In others, it is correct to have either a positive or negative value. The sign helps determine whether recorded values consistently fall above or below expected values.

Percent Error Formula

Percent error is the difference between a measured or experiment value and an accepted or known value, divided by the known value, multiplied by 100%.

For many applications, percent error is always expressed as a positive value. The absolute value of the error is divided by an accepted value and given as a percent.

Percent Error = | Accepted Value - Experimental Value | / Accepted Value x 100%

For chemistry and other sciences, it is customary to keep a negative value, should one occur. Whether error is positive or negative is important. For example, you would not expect to have a positive percent error comparing actual to theoretical yield in a chemical reaction . If a positive value was calculated, this would give clues as to potential problems with the procedure or unaccounted reactions.

When keeping the sign for error, the calculation is the experimental or measured value minus the known or theoretical value, divided by the theoretical value and multiplied by 100%.

Percent Error = [Experimental Value - Theoretical Value] / Theoretical Value x 100%

Percent Error Calculation Steps

• Subtract one value from another. The order does not matter if you are dropping the sign (taking the absolute value. Subtract the theoretical value from the experimental value if you are keeping negative signs. This value is your "error."
• Divide the error by the exact or ideal value (not your experimental or measured value). This will yield a decimal number.
• Convert the decimal number into a percentage by multiplying it by 100.
• Add a percent or % symbol to report your percent error value.

Percent Error Example Calculation

In a lab, you are given a block of aluminum . You measure the dimensions of the block and its displacement in a container of a known volume of water. You calculate the density of the block of aluminum to be 2.68 g/cm 3 . You look up the density of a block of aluminum at room temperature and find it to be 2.70 g/cm 3 . Calculate the percent error of your measurement.

• Subtract one value from the other: 2.68 - 2.70 = -0.02
• Depending on what you need, you may discard any negative sign (take the absolute value): 0.02 This is the error.
• Divide the error by the true value: 0.02/2.70 = 0.0074074
• Multiply this value by 100% to obtain the percent error: 0.0074074 x 100% = 0.74% (expressed using two significant figures ). Significant figures are important in science. If you report an answer using too many or too few, it may be considered incorrect, even if you set up the problem properly.

Percent Error vs. Absolute and Relative Error

Percent error is related to absolute error and relative error . The difference between an experimental and known value is the absolute error. When you divide that number by the known value you get relative error . Percent error is relative error multiplied by 100%. In all cases, report values using the appropriate number of significant digits.

Why Is Percent Error Important?

Percent error is used extensively across various fields such as physics, chemistry, engineering, and statistics. Because it measures deviations from a true value or accepted value, percent error can be utilized to validate hypotheses during experiments or ensure quality control in manufacturing processes.

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What distinguishes experimental value from theoretical value?

Experimental values and theoretical values are two different types of values used in scientific research and analysis. Here's how they can be distinguished:

Experimental Values:

• Experimental values are obtained through actual experiments or observations [1] .
• They are measured or observed quantities that are directly obtained from experiments conducted in a laboratory or in the field.
• Experimental values are subject to uncertainties and errors due to various factors such as limitations in measurement equipment, experimental procedures, and environmental conditions.
• These values are specific to the particular experiment or observation and may vary from one experiment to another.
• Experimental values are used to validate or test hypotheses, theories, or models.

Theoretical Values:

• Theoretical values are calculated based on established theories, equations, or models [1] .
• They are derived from mathematical or computational models that describe the behavior or properties of a system.
• Theoretical values are not obtained through direct measurement or observation but are instead predicted or estimated based on existing knowledge and understanding.
• These values are often used as reference points or benchmarks for comparison with experimental values.
• Theoretical values are generally considered to be ideal or idealized representations of the system under study.

Relationship between Experimental and Theoretical Values:

• Experimental values can be compared to theoretical values to assess the accuracy and validity of a theory or equation [2] .
• If the experimental values closely match the theoretical values, it provides support for the theory or equation being tested.
• Discrepancies between experimental and theoretical values can indicate the presence of experimental errors, limitations in the theoretical model, or the need for further investigation.

Learn more:

• What factors can contributed for the experimental value to be ...
• Experimental Values vs. Theoretical Values
• Percent Error and Percent Difference

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What is the difference between Accepted Value vs. Experimental Value?

#"Error" = "|experimental value - accepted value|"#

The difference is usually expressed as percent error .

#"% error" = "|experimental value - accepted value|"/"experimental value" × 100 %#

For example, suppose that you did an experiment to determine the boiling point of water and got a value of 99.3 °C.

Your experimental value is 99.3 °C.

The theoretical value is 100.0 °C.

The experimental error is #"|99.3 °C - 100.0 °C| = 0.7 °C"#

The percent error is #"|99.3 °C - 100.0 °C|"/"100.0 °C" = "0.7 °C"/"100.0 °C" × 100% = 0.7 %#

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How to calculate theoretical and experimental data in general physics´experiments?

Im struggling how to calculate theoretical and experimental data with the added formulas and second Newton law. I did a free body diagram but it doesnt clarify how to calculate it. Any suggestion will be welcome. thank you.

The experimental data is just the measurement of the time to reach each sensor, so the two lines above the chart. You should copy the times into the second row of the chart. From the inclination of the plane you should be able to calculate a predicted acceleration due to gravity. Presumably you start with the ball at rest, so $v_0=0$ . You can then predict the velocity as a function of time from your equation, the time the ball should have passed each sensor, and compare that with the measured data. I am not sure how your professor expects you to come up with the experimental values of velocity and acceleration. It might be an overall fit to get the acceleration. It might be computing the change in distance divided by the change in time, but that has the problem that the velocity is constantly changing.

• $\begingroup$ Thanks. So I think that my teacher wants to complete the chart with the added formulas. I can do it but, I need to clarify what is theoretical and what experimental. It seems confusing at first sight. $\endgroup$ –  Charlie Van Basten Øydne Commented Aug 13, 2020 at 1:35

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Determine acceleration from experiment (Newton 2nd Law)

I have done a physics experiment (setup below). And was asked to determine the experimental and theoretical acceleration.

The data I've got

Ok, am I right to say

Experimental acceleration = $2(s_f - s_i) / t^2$

Theoratical acceleration = $m_2 \times 0.98 / m_1$

Percentage discrepancy = $\frac{|(Experimental - Theoretical)|}{Theoretical} \times 100$%

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• 1 $\begingroup$ Hi Jiew - your question seems kind of unfocused. What concept is it specifically that is confusing you? Are you confused about why the mass of the cart affects its horizontal acceleration? Or are you confused about some step in the calculation of the theoretical acceleration? It would help a lot if you edit your question to focus on the one thing you want to ask. If you have multiple concepts to ask about, you can post more than one question about the same lab setup. $\endgroup$ –  David Z Commented Sep 27, 2012 at 4:30
• $\begingroup$ Ok, I edited my post and posted another question physics.stackexchange.com/questions/38448/… $\endgroup$ –  Jiew Meng Commented Sep 27, 2012 at 6:25

If by $s_f$ and $s_i$ you mean the final and initial position, respectively --- so that $s_f-s_i$ is just $d$ in your table --- then yes, your experimental acceleration is right. As for your theoretical acceleration, it should be $\frac{9.8m_2}{m_1}$, not $0.98$ --- the acceleration due to gravity is $g=9.8$. I'm assuming you just made a typo. Your definition of percentage discrepancy is right.

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Experimental Values vs. Theoretical Values

• Thread starter DBJKIBA
• Start date Sep 30, 2009
• Tags Experimental Theoretical
• Sep 30, 2009
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A PF SuperCluster

Depends on how you are determining the speed of sound.  

• Oct 1, 2009

A PF Planet

Hi DBJKIBA- Many important experiments were successful BECAUSE the experimenters did not believe the theoretical value was correct. In fact it wasn't. I recall a theoretical calculation of the electron g factor (by Karplus and Kroll) that was wrong, and the disparity with previous experiments led to a succession of new and better experiments, and a recalculation of g. Bob S  

Related to Experimental Values vs. Theoretical Values

1. what is the difference between experimental values and theoretical values.

Experimental values are the values obtained through actual experiments or observations, while theoretical values are calculated based on established theories or equations.

2. How do experimental values and theoretical values relate to each other?

Experimental values can be compared to theoretical values to assess the accuracy and validity of a theory or equation. If the experimental values closely match the theoretical values, it provides support for the theory.

3. Why do experimental values sometimes differ from theoretical values?

This can be due to a variety of factors, such as experimental error, limitations in measurement equipment, or unforeseen variables that affect the results. It is important to carefully analyze and evaluate any discrepancies between experimental and theoretical values.

4. How can experimental values be improved to better match theoretical values?

To improve the accuracy of experimental values, scientists can use more precise measurement equipment, conduct multiple trials, and carefully control variables in the experiment. It is also important to critically assess and refine the experimental methods being used.

5. Can experimental values ever be exactly the same as theoretical values?

In most cases, there will always be some degree of difference between experimental and theoretical values due to the inherent uncertainties and complexities of the natural world. However, by continually refining and improving experimental methods, scientists can strive to minimize these differences and increase the accuracy of their results.

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Theoretical Particle Physics Overview

The Standard Model (SM) of strong, electromagnetic and weak interactions is the crowning achievement of twentieth century physics. However, despite its many spectacular successes, the SM is theoretically inconsistent at high energies and should be superseded by a new, more fundamental theory at the teraelectron-volt (TeV) energy scale. In addition, the SM cannot incorporate dark matter, whose existence has been confirmed by numerous astrophysical observations.

Many theoretical ideas about the physics at the TeV scale and the nature of dark matter have been proposed; examples include supersymmetry, extra dimensions of space, and new strong interactions. Members of Cornell theory group are active in investigating these ideas and their experimental and observational consequences. Currently, the Large Hadron Collider (LHC) at the CERN laboratory in Switzerland is exploring the TeV scale experimentally for the first time in history. Theoretical interpretation of the LHC data is expected to be a major focus of research in the next few years. In this work, Cornell theorists benefit from traditionally close connections with the LEPP experimental group , which participates in the CMS experiment at the LHC.

Another research area actively pursued at Cornell is string theory, which combines quantum field theory and gravity in a consistent framework. A key goal is to understand the properties of the four-dimensional effective theories derived from compactifications of string theory. Cornell theorists create new analytical techniques for the study of flux compactifications, use these tools to find novel solutions of supergravity, and then characterize the resulting effective actions. A primary application of these methods is in the study of the very early universe: questions about inflation can often be mapped into questions about the geometry of the internal space or about the potential governing deformations of this space. Theorists at Cornell have led the exploration of the interface between string theory and inflationary cosmology, which holds the prospect of bringing string theory into contact with cosmological observations.

Particle theory students at Cornell have the opportunity to explore a wide range of research areas, ranging from experiment-driven theory to highly mathematical analyses of supersymmetric field theories or quantum theories of gravity. There is also work at the interface between condensed matter physics and particle physics, where mathematical and numerical techniques from relativistic quantum field theory are adapted for use on condensed matter systems, and ideas from condensed matter physics are applied to quantum field theories. Work can be analytical, or it can be computational, as in numerical simulations of quantum chromodynamics and other quantum field theories—a research area invented at Cornell. The particle theory program is very flexible: it is easy for theory students to work in more than one area, and it is not unusual for a student to co-author papers with more than one professor during their graduate career. In addition, there is close collaboration with Cornell’s theoretical astrophysics group, focused on problems of common interest to particle physics and astrophysics/cosmology.

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Effective removal of cu(ii) ions from aqueous solution by cross-linked chitosan-based hydrogels.

1. Introduction

2. experimental section, 2.1. materials, 2.2. ph value of zero-point charge (phzpc), 2.3. adsorption investigations, 2.4. adsorption kinetic studies.

• Pseudo-first-order model
• Pseudo-second-order model
• Elovich model
• Intraparticle diffusion model

2.5. Investigation of Adsorption Isotherms

• The Langmuir isotherm model is expressed by Equations (12)–(14):
• The Freundlich isotherm model is expressed by Equations (15) and (16):
• Temkin isotherm model
• Dubinin–Radushkevich (D-R) isotherm model

2.6. Desorption Investigation

3. results and discussion, 3.1. ph of zero-point charge (phzpc), 3.2. optimizing adsorption, 3.2.1. effect of temperature, 3.2.2. effect of the ph, 3.2.3. hydrogel dosage effect, 3.2.4. cross-linking content effect, 3.3. adsorption kinetics, 3.4. isotherms of adsorption, 3.5. desorption evaluation, 3.6. comparison of adsorption capacity of different adsorbents for cu(ii) ions, 4. conclusions, supplementary materials, author contributions, data availability statement, acknowledgments, conflicts of interest.

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Click here to enlarge figure

Samples	Trimellitic Anhydride Chloride (mmol)	Ammonium Thiocyanate (mmol)	Chitosan (mmol)	Elemental Analysis (%)
				C	H	N	O	S
Chitosan	-	-	-	45.10	6.77	8.43	39.70	-
H	10.0	10.0	40	46.61	5.48	7.86	36.81	3.24
H	20.0	20.0	40	47.54	4.68	7.51	35.03	5.24

Kinetic Models	Parameters	H	H
q (mg g )		96.2	97.59
Pseudo- first-order	R	0.645	0.944
	q (mg g )	3.043	1.98
	k (min )	0.0151	0.0166
	Δq	47.14	52.23
Pseudo- second-order	R	0.999	1
	q (mg g )	98.04	99.01
	k (10 )	0.005	0.0146
	(g mg min )
	Δq	3.2	2.93
Elovich	R	0.722	0.979
	β (g mg )	0.361	0.698
	α (10 )	2.714	1.105 × 10
	(mg g min )
	Δq	85.44	42.64
Intraparticle diffusion	R	0.778	0.967
	k (mg g min	0.853	0.446
	Δq	89.79	93.47

Models	Parameter	H	H
Langmuir	q (mg g )	2500	227.27
	R	0.333–0.833	0.357–0.833
	K (L mg )	0.002	0.068
	R	0.014	0.744
Freundlich	1/n	0.969	0.681
	K (mg g )	6.33	18.05
	R	0.965	0.952
Temkin	B (kJ mol )	62.7	44.02
	K (L g )	2.921	1.066
	R	0.802	0.843
D-R	q (mg g )	113.52	101.19
	E kJ mol	0.408	0.845
	B × 10	3	0.7
	R	0.785	0.765

Adsorbent	Adsorption Capacity (mg g )	Temperature (°C)	Metal Conc. (mg L )	Adsorbent Dose (g)	pH	Ref.
Cross-linked chitosan grafted with polyaniline	131.58	20–40	100	0.05	6	[ ]
Quaternized chitosan@chitosan cationic polyelectrolyte microsphere	687.6	25	0–2000	0.075	5	[ ]
Horn Core calcined at 400 °C (P400)	99.98	25	100–500	0.02	5	[ ]
Magnetic chitosan@bismuth tungstate coated by silver (MCTS-Ag/ Bi WO )	181.8	20–40	20 -120	0.02	6	[ ]
Xanthate-modified magnetic chitosan	34.5	25	100	-	5	[ ]
Epichlorohydrin cross-linked xanthate chitosan	43.47	50	100	-	5	[ ]
Medicinal plant (Salvadora persica)	74.30	25	100	-	4	[ ]
Pecan nutshell (Carya illinoinensis)	23.37	30	-	-	5	[ ]
H	96.20	25	100	0.01	6	Present study
H	97.59	25	100	0.01	6	Present study

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Al-Harby, N.F.; Alrasheedi, M.; Mohammed, A.e.M.E.; Soliman, S.M.A.; Mohamed, N.A. Effective Removal of Cu(II) Ions from Aqueous Solution by Cross-Linked Chitosan-Based Hydrogels. Water 2024 , 16 , 2324. https://doi.org/10.3390/w16162324

Al-Harby NF, Alrasheedi M, Mohammed AeME, Soliman SMA, Mohamed NA. Effective Removal of Cu(II) Ions from Aqueous Solution by Cross-Linked Chitosan-Based Hydrogels. Water . 2024; 16(16):2324. https://doi.org/10.3390/w16162324

Al-Harby, Nouf F., Muneera Alrasheedi, Ard elshifa M. E. Mohammed, Soliman M. A. Soliman, and Nadia A. Mohamed. 2024. "Effective Removal of Cu(II) Ions from Aqueous Solution by Cross-Linked Chitosan-Based Hydrogels" Water 16, no. 16: 2324. https://doi.org/10.3390/w16162324

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Computer Science > Computers and Society

Title: personhood credentials: artificial intelligence and the value of privacy-preserving tools to distinguish who is real online.

Abstract: Anonymity is an important principle online. However, malicious actors have long used misleading identities to conduct fraud, spread disinformation, and carry out other deceptive schemes. With the advent of increasingly capable AI, bad actors can amplify the potential scale and effectiveness of their operations, intensifying the challenge of balancing anonymity and trustworthiness online. In this paper, we analyze the value of a new tool to address this challenge: "personhood credentials" (PHCs), digital credentials that empower users to demonstrate that they are real people -- not AIs -- to online services, without disclosing any personal information. Such credentials can be issued by a range of trusted institutions -- governments or otherwise. A PHC system, according to our definition, could be local or global, and does not need to be biometrics-based. Two trends in AI contribute to the urgency of the challenge: AI's increasing indistinguishability (i.e., lifelike content and avatars, agentic activity) from people online, and AI's increasing scalability (i.e., cost-effectiveness, accessibility). Drawing on a long history of research into anonymous credentials and "proof-of-personhood" systems, personhood credentials give people a way to signal their trustworthiness on online platforms, and offer service providers new tools for reducing misuse by bad actors. In contrast, existing countermeasures to automated deception -- such as CAPTCHAs -- are inadequate against sophisticated AI, while stringent identity verification solutions are insufficiently private for many use-cases. After surveying the benefits of personhood credentials, we also examine deployment risks and design challenges. We conclude with actionable next steps for policymakers, technologists, and standards bodies to consider in consultation with the public.

Comments:	63 pages, 7 figures, 5 tables
Subjects:	Computers and Society (cs.CY)
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