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Theoretical vs. Experimental Probability

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Theoretical Probability versus Experimental Probability

You've heard the terms, theoretical probability and experimental probability , but what do they mean?

Are they in anyway related? This is what we are going to discover in this lesson.

If you've completed the lessons on i ndependent and dependent probability , then you've already found the theoretical probability for numerous problems.

Theoretical Probability

Theoretical probability is the probability that is calculated using math formulas. This is the probability based on math theory.

Experimental Probability

Experimental probability is calculated when the actual situation or problem is performed as an experiment. In this case, you would perform the experiment, and use the actual results to determine the probability.

In order to accurately perform an experiment, you must:

Identify what constitutes a " trial ".
Perform a minimum of 25 trials
Set up an organizer (table or chart) to record your data.

Let's take a look at an example where we first calculate the theoretical probability, and then perform the experiment to determine the experimental probability.

It will be interesting to compare the theoretical probability and the experimental probability. Do you think the two calculations will be close?

Example 1 - Theoretical Versus Experimental

This problem is from Example 1 in the independent events lesson. We calculated the theoretical probability to be 1/12 or 8.3%. Take a look:

Since we know that the theoretical probability is 8.3% chance of flipping a head and rolling a 6, let's see what happens when we actually perform the experiment.

Identify a trial: A trial consists of flipping a coin once and rolling a die once.

Conduct 25 trials and record your data in the table below.

For each trial, I flipped the coin once and rolled the die. I recorded and H for heads and a T for tails in the row labeled "Coin."

I recorded the number on the die in the row labeled "Die".

In the last row I determined whether the trial completed the event of flipping a head and rolling a six.

In this experiment, there was only 1 trial (out of 25) where a head was flipped on the coin and a 6 was rolled on the die.

This means that the experimental probability is 1/25 or 4%.

Please note that everyone's experiment will be different; thus allowing the experimental probability to differ.

Also, the more trials that you conduct in your experiment, the closer your calculations will be for the experimental and theoretical probabilities.

Conclusions

The theoretical probability is 8.3% and the experimental probability is 4%. Although the experimental probability is slightly lower, this is not a significant difference.

In most experiments, the theoretical probability and experimental probability will not be equal; however, they should be relatively close.

If the calculations are not close, then there's a possibility that the experiment was conducted improperly or more trials need to be completed.

I hope this helps to give you a sense of how to set up an experiment in order to compare theoretical versus experimental probabilities.

Probability
Theoretical/Experimental Probability

why is experimental and theoretical probability different

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Experimental Probability vs. Theoretical Probability

What's the difference.

Experimental probability is based on actual observations and data collected from experiments or real-life events. It is calculated by dividing the number of favorable outcomes by the total number of outcomes. On the other hand, theoretical probability is based on mathematical calculations and predictions. It is determined by analyzing the possible outcomes and their likelihood in a given situation. While experimental probability provides an estimate of the likelihood of an event based on real-world data, theoretical probability provides a more precise and accurate prediction based on mathematical principles.

Further Detail

Introduction.

Probability is a fundamental concept in mathematics and statistics that allows us to quantify the likelihood of an event occurring. It plays a crucial role in various fields, including science, finance, and everyday decision-making. When discussing probability, two important terms often come up: experimental probability and theoretical probability. While both concepts deal with the likelihood of events, they differ in their approach and application. In this article, we will explore the attributes of experimental probability and theoretical probability, highlighting their similarities and differences.

Experimental Probability

Experimental probability, also known as empirical probability, is based on observations and data collected from experiments or real-life events. It involves conducting experiments or observations to determine the likelihood of an event occurring. The experimental probability of an event is calculated by dividing the number of times the event occurs by the total number of trials or observations.

One of the key attributes of experimental probability is its reliance on real-world data. By conducting experiments or observations, we can gather empirical evidence to estimate the probability of an event. This makes experimental probability particularly useful when dealing with situations where theoretical calculations may be challenging or impractical.

Another attribute of experimental probability is its subjectivity. Since it is based on observed data, the results can vary depending on the specific experiments or observations conducted. The more trials or observations we perform, the more reliable the experimental probability becomes. However, it is important to note that experimental probability is still an estimation and may not always accurately reflect the true probability of an event.

Experimental probability is often used in fields such as psychology, biology, and social sciences, where controlled experiments or observations can provide valuable insights into the likelihood of certain outcomes. For example, in a psychology study, researchers may conduct experiments to determine the probability of a specific behavior occurring in response to certain stimuli.

Theoretical Probability

Theoretical probability, also known as classical probability, is based on mathematical principles and calculations. It involves analyzing the underlying structure of a given situation or event to determine the probability of specific outcomes. Theoretical probability relies on assumptions and mathematical models to make predictions about the likelihood of events.

One of the key attributes of theoretical probability is its objectivity. Since it is based on mathematical calculations, the results are not influenced by specific experiments or observations. Theoretical probability provides a systematic and consistent approach to quantifying probabilities, making it particularly useful in situations where empirical data may be limited or unavailable.

Another attribute of theoretical probability is its precision. By using mathematical formulas and principles, we can calculate the exact probability of an event occurring. This allows for precise predictions and analysis, which can be valuable in fields such as finance, engineering, and physics.

Theoretical probability is often used in situations where the outcomes are well-defined and the underlying probabilities can be determined with certainty. For example, in a fair six-sided die, the theoretical probability of rolling a specific number is 1/6, as there are six equally likely outcomes.

While experimental probability and theoretical probability differ in their approach and application, they share some common attributes. Both concepts deal with the likelihood of events and aim to quantify probabilities. Additionally, both experimental and theoretical probabilities range from 0 to 1, where 0 represents an impossible event and 1 represents a certain event.

However, there are also notable differences between experimental and theoretical probability. Experimental probability relies on observed data, making it subjective and dependent on the specific experiments or observations conducted. On the other hand, theoretical probability is objective and based on mathematical calculations, providing a more systematic and consistent approach.

Another difference lies in the precision of the probabilities. Experimental probability provides an estimation of the likelihood of an event based on observed data, which may not always accurately reflect the true probability. Theoretical probability, on the other hand, allows for precise calculations and predictions, assuming the underlying assumptions and mathematical models are accurate.

Furthermore, experimental probability is often used in situations where real-world data is available or when conducting experiments is feasible. It is particularly useful in fields such as social sciences, where controlled experiments or observations can provide valuable insights. Theoretical probability, on the other hand, is more suitable for situations where the underlying probabilities can be determined with certainty or when empirical data is limited or unavailable. It is commonly used in fields such as mathematics, finance, and physics.

Experimental probability and theoretical probability are two important concepts in the study of probability. While both aim to quantify the likelihood of events, they differ in their approach and application. Experimental probability relies on observed data and provides an estimation of probabilities based on experiments or observations. Theoretical probability, on the other hand, is based on mathematical calculations and provides precise predictions assuming the underlying assumptions and models are accurate.

Understanding the attributes of experimental and theoretical probability is crucial for making informed decisions and analyzing probabilities in various fields. By recognizing the strengths and limitations of each approach, we can effectively apply probability concepts to real-world situations and enhance our understanding of uncertain events.

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